Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #139624 |
| commutator | #379 | C22×C4 |
| Frattini, cosocle | #2309 | C23 |
| prime radical | C22×H(5,𝔽2) | D4 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1219 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 1219 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,5,10,7,12,22,35)(3,13,15,39,17,40,44,56)(4,9,18,30,20,32,49,60)(6,11,23,36,24,37,21,34)(8,25,27,16,28,45,33,46)(14,38,41,63,43,55,47,64)(19,31,51,61,52,62,48,59)(26,53,54,42,29,57,58,50);(1,3,7,17)(2,8)(4,14,20,43)(5,15,22,44)(6,16,24,46)(9,29)(10,33)(11,13)(12,28)(18,47,49,41)(19,50,52,42)(21,25,23,45)(26,32)(27,35)(30,58)(31,55)(34,39)(36,56)(37,40)(38,62)(48,53,51,57)(54,60)(59,63)(61,64);(1,4,7,20)(2,9,12,32)(3,14,17,43)(5,18,22,49)(6,19)(8,26)(10,30,35,60)(11,31)(13,38,40,55)(15,41,44,47)(16,42)(21,48)(23,51)(24,52)(25,53)(27,54)(28,29)(33,58)(34,59)(36,61)(37,62)(39,63,56,64)(45,57)(46,50)Polycyclic presentation:
2^10|a^2=d,b^2=ij,c^2=ghij,d^2=j,e^2=gij,b^a=be,c^b=cf,d^b=dg,e^a=eij,e^b=egi,e^c=eh,e^d=ej,f^a=fhij,f^c=fi,f^d=fj,f^e=fij,g^a=gj,g^c=gj,h^b=hi,h^c=hj,h^e=hj,i^a=ij55 conjugacy classes — view the full character table.
| maximal subgroup of | #72285553, #72285694, #72285726, … |
| maximal normal subgroup of | #72285553, #72285694, #72285726, … |
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.4);;
gap> SetPower(coll, 2, F.9*F.10);;
gap> SetPower(coll, 3, F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*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.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);