Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #152892 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | C22×C32 | C23 |
| prime radical | #513054 | C2 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ #13549 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(1,2,5,11,31,27,80,86,8,12,33,50,56,41,102,103,37,54,97,113,114,99,127,126,98,89,112,94,35,29,79,88,6,14,32,52,95,90,124,125,36,55,96,93,39,28,81,87,9,13,34,51,53,40,100,92,38,26,49,30,7,4,23,25)(3,15,18,59,67,64,117,119,21,60,73,91,108,78,107,83,75,48,71,44,104,109,128,120,115,110,111,66,57,65,69,62,19,47,17,58,68,101,121,118,74,85,70,116,43,24,82,84,22,10,42,45,105,77,123,122,76,46,106,72,20,16,61,63);(1,3,6,19)(2,10)(4,24)(5,17,32,18)(7,20,35,57)(8,21,36,74)(9,22,37,75)(11,44)(12,46)(13,47)(14,48)(15,54)(16,41)(23,69,79,61)(25,83)(26,85)(27,77)(28,65)(29,78)(30,91)(31,67,95,68)(33,70,96,73)(34,71,97,42)(38,76,98,115)(39,43,56,108)(40,101)(45,52)(49,111,112,106)(50,66)(51,59)(53,105,114,104)(55,110)(58,113)(60,89)(62,103)(63,87)(64,99)(72,93)(80,121,124,117)(81,107,102,82)(84,88)(86,120)(90,109)(92,119)(94,116)(100,128,127,123)(118,126)(122,125);(1,4)(2,7)(3,16,19,65)(5,30)(6,29)(8,40)(9,41)(10,43,48,108)(11,49)(12,53)(13,56)(14,35)(15,57,47,20)(17,66,18,72)(21,77,74,109)(22,78,75,24)(23,25)(26,31)(27,38)(28,37)(32,94)(33,51)(34,50)(36,99)(39,54)(42,91,71,116)(44,73,45,70)(46,68,110,67)(52,112)(55,114)(58,106,59,111)(60,104,85,105)(61,63,69,62)(64,115,101,76)(79,88)(80,92)(81,103)(82,83,107,84)(86,100)(87,102)(89,95)(90,98)(93,97)(96,113)(117,122,121,120)(118,123,119,128)(124,126)(125,127)Polycyclic presentation:
2^10|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bij,c^a=cd,c^b=cfi,d^c=dg,e^b=ej,e^c=eg,f^a=fj,f^c=fj,g^c=gh,h^c=hi,i^c=ij100 conjugacy classes — view the full character table.
| maximal subgroup of | #72322773, #72322926, #72322965 |
| maximal normal subgroup of | #72322773, #72322926, #72322965 |
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, 2, F.6);;
gap> SetPower(coll, 3, F.4*F.5*F.7);;
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.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
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);