Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×#3573 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4⋊#1020 |
| commutator, cosocle | #5950 | C23 |
| Frattini | C4 | C22×#1020 |
| Fitting | C2×#3573 | D3 |
| socle | C2 | C2×C4⋊#1020 |
| hypercenter | C2×C8 | #1020 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ #5950 ▷ #997 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
2809 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 2809 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,3,14,36,7,18,4,15)(2,9,19,33,8,25,10,13)(5,43,58,56,6,60,20,57,22,16,21,64,23,17,46,39,24,41,37,40,47,42,38,59)(11,49,61,53,12,62,30,54,32,28,31,63,26,29,52,34,27,51,44,35,50,48,45,55);(1,5)(2,26)(3,16)(4,20)(6,22)(7,23)(8,11)(9,48)(10,44)(12,50)(13,53)(14,37)(15,39)(17,41)(18,42)(19,30)(21,46)(24,47)(25,28)(27,32)(29,62)(31,61)(33,34)(35,63)(36,56)(38,58)(40,59)(43,60)(45,52)(49,51)(54,55)(57,64);(1,2)(3,13)(4,19)(5,12)(6,11)(7,8)(9,15)(10,14)(16,35)(17,34)(18,33)(20,45)(21,44)(22,32)(23,27)(24,26)(25,36)(28,40)(29,39)(30,38)(31,37)(41,55)(42,54)(43,53)(46,61)(47,50)(48,57)(49,56)(51,59)(52,58)(60,63)(62,64)Polycyclic presentation:
2^4,3,2^5|c^2=dj,d^2=j,b^a=bj,c^a=cd,d^a=dj,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341802, #72341861, #72341879, … |
| maximal normal subgroup of | #72341802, #72341861, #72341879, … |
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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.4*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);