Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #151068 |
| commutator | C2×C32 | C22×C4 |
| Frattini, cosocle | #1397 | C23 |
| prime radical | C23 | #1426 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ C4×M64 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 75 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)(3,15,19,60,16,57,40,62)(4,22,23,77)(5,29)(6,35)(7,39)(8,9)(10,18,45,71,41,63,56,73)(11,26,42,83)(12,32)(13,38)(14,37)(17,53)(20,75,74,124,64,126,103,120)(21,76,36,61,65,116,66,123)(24,85,86,112,81,59,89,52)(25,54)(27,94,67,118)(28,97,84,80)(30,48,102,115,101,110,70,117)(31,44)(33,49)(34,98)(43,68)(46,88,114,127,108,128,122,95)(47,106,55,72,109,87,113,125)(50,119,111,90)(51,121,107,93)(58,96,79,91)(69,105,92,78)(82,99)(100,104);(2,9)(3,16)(4,23)(12,43)(13,44)(14,39)(15,57)(19,40)(20,64)(21,65)(26,83)(27,67)(28,84)(29,98)(30,101)(35,53)(36,66)(48,110)(49,100)(54,99)(60,62)(61,123)(69,92)(70,102)(74,103)(75,126)(76,116)(90,119)(91,96)(93,121)(115,117)(120,124);(1,3,17,66,37,19,38,21,8,16,6,36,7,40,31,65)(2,10,44,113,39,45,35,47,9,41,13,55,14,56,53,109)(4,24,27,95,84,89,91,128,23,81,67,127,28,86,96,88)(5,30,68,64,104,102,82,103,34,101,32,20,33,70,25,74)(11,48,50,120,107,117,78,126,42,110,111,124,51,115,105,75)(12,52,98,108,99,85,100,122,43,112,29,46,54,59,49,114)(15,58,76,80,62,94,123,22,57,79,116,97,60,118,61,77)(18,69,106,93,73,119,125,26,63,92,87,121,71,90,72,83)Polycyclic presentation:
2^10|a^2=e,c^2=ehj,d^2=fhij,e^2=g,f^2=hij,h^2=ij,i^2=j,b^a=bgj,c^a=cd,c^b=cj,d^a=df,d^b=dj,d^c=df,e^c=eij,e^d=ej,f^a=fh,f^c=fh,g^c=gj,h^a=hi,h^c=hi,i^a=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72321944, #72321964, #72321979 |
| maximal normal subgroup of | #72321944, #72321964, #72321979 |
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.5*F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);