Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×#1020 |
| commutator, hypocenter | #997 | C2×C8 |
| Fitting | C2×#3573 | D3 |
| socle | C2 | C2×#85561 |
| prime radical | C2×#5950 | C4 |
| cosocle | #85666 | C22 |
| hypercenter | C2×C8 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
1427 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 1427 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 2 generators on 64 points:
(2,8)(5,22,20)(6,42,21)(9,23)(10,24)(11,12,32,25,26,48)(14,38,36)(15,56,37)(17,41,39)(18,57,40)(27,43)(28,29,51,44,45,60)(30,31,52,46,47,61)(33,55,53)(34,63,54)(49,50,62,58,59,64);(1,11,36,29,4,30,53,50,7,25,56,45,19,46,63,59)(2,14,51,18,10,33,62,22,8,37,60,41,24,54,64,6)(3,28,39,31,13,49,42,26,16,44,57,47,35,58,20,12)(5,32,15,9,17,52,34,27,21,48,38,23,40,61,55,43)Polycyclic presentation:
2^4,3,2^5|a^2=c,c^2=d,d^2=j,b^a=bj,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=ij92 conjugacy classes — view the full character table.
| maximal subgroup of | #72342051 |
| maximal normal subgroup of | #72342051 |
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, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*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);