Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152629 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | C22×C32 | C23 |
| prime radical | #512945 | C2 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ C2×#2258 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
381 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,13,33,27,81,83,9,14,39,55,53,46,97,101,42,58,37,89,57,99,113,98,52,85,87,38,12,29,35,80,7,16,5,30,34,86,95,127,41,59,36,88,44,28,82,84,10,15,40,56,96,45,100,94,43,26,54,32,8,4,24,25)(3,17,19,47,70,103,122,117,22,61,72,104,110,68,116,120,75,50,93,108,112,78,125,126,111,66,106,90,51,18,63,65,20,11,31,48,91,67,115,119,74,49,92,107,77,79,123,118,23,62,73,105,109,124,128,121,76,114,102,69,21,60,71,64);(1,3)(2,11)(4,18)(5,31)(6,19)(7,20)(8,21)(9,22)(10,23)(12,51)(13,48)(14,49)(15,50)(16,17)(24,71)(25,65)(26,66)(27,67)(28,68)(29,60)(30,47)(32,90)(33,70)(34,91)(35,63)(36,92)(37,93)(38,69)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(52,111)(53,110)(54,102)(55,107)(56,108)(57,112)(58,62)(59,61)(64,80)(81,122)(82,123)(83,119)(84,120)(85,114)(86,103)(87,106)(88,104)(89,105)(94,126)(95,115)(96,109)(97,116)(98,121)(99,124)(100,128)(101,118)(113,125)(117,127);(1,4,7,29)(2,12,16,8)(3,18)(5,32,6,38)(9,45,41,99)(10,46,42,28)(11,21)(13,54,30,87)(14,57,59,96)(15,44,58,53)(17,51)(19,69)(20,60)(22,78)(23,79)(24,80,35,25)(26,34,85,33)(27,52,86,43)(31,90)(36,56,39,89)(37,55,40,88)(47,102)(48,106)(49,109)(50,110)(61,112)(62,77)(63,65)(64,71)(66,70)(67,76)(68,75)(72,105)(73,104)(74,124)(81,98,95,94)(82,84,97,101)(83,100,127,113)(91,114)(92,108)(93,107)(103,111)(115,126)(116,118)(117,128)(119,125)(120,123)(121,122)Polycyclic presentation:
2^10|a^2=e,c^2=defgj,d^2=gh,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bj,c^a=cd,c^b=cf,d^a=dj,d^b=dj,d^c=dg,e^c=egj,f^a=fj,f^c=fj,g^c=gh,h^c=hi,i^c=ij148 conjugacy classes — view the full character table.
| maximal subgroup of | #72322773, #72322901, #72322972 |
| maximal normal subgroup of | #72322773, #72322901, #72322972 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.4*F.5*F.6*F.7*F.10);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);