Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #130062 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #2280 | C23 |
| prime radical | #23895 | C4 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #2280 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1122 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2)(3,8)(4,19)(5,10)(6,11)(7,13)(9,22)(12,26)(14,33)(15,41)(16,31)(17,30)(18,32)(20,47)(21,34)(23,48)(24,36)(25,38)(27,39)(28,40)(29,43)(35,51)(37,52)(42,59)(44,57)(45,58)(46,55)(49,63)(50,62)(53,60)(54,61)(56,64);(1,3)(2,8)(4,20)(5,17)(6,27)(7,18)(9,33)(10,30)(11,39)(12,41)(13,32)(14,22)(15,26)(16,28)(19,47)(21,42)(23,49)(24,43)(25,46)(29,36)(31,40)(34,59)(35,61)(37,58)(38,55)(44,50)(45,52)(48,63)(51,54)(53,64)(56,60)(57,62);(1,4,21,50,7,24,25,51)(2,9,34,60,13,37,38,48)(3,14,42,64,18,45,46,49)(5,22,28,53,26,52,6,23)(8,29,55,57,32,47,59,61)(10,19,40,62,12,36,11,35)(15,43,27,54,17,20,16,44)(30,33,39,63,41,58,31,56)Polycyclic presentation:
2^10|c^2=dehj,d^2=h,e^2=hj,f^2=hij,c^a=cd,c^b=cf,d^a=dh,d^b=dg,e^b=eghij,f^a=fgij,f^b=fhij,f^d=fij,f^e=fij,g^a=gj,g^c=gi,g^d=gj,g^e=gj,g^f=gj,h^b=hj,i^a=ij,i^b=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72285435, #72285567 |
| maximal normal subgroup of | #72285435, #72285567 |
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, 3, F.4*F.5*F.8*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.8*F.9*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.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);