Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #20609 |
| commutator | C22×D4 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #7743195 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
5561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(11,19)(17,33)(20,34)(21,36)(22,25)(31,35)(37,40)(38,39);(1,3)(2,5,6,7)(4,8)(9,10)(11,20)(12,16,24,32)(13,17)(14,18)(15,22)(19,34)(21,37,36,40)(23,31,26,35)(25,29)(27,33)(28,38,30,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,19)(10,15)(12,21,24,36)(13,25,27,22)(16,30)(17,34)(18,29)(20,33)(23,39,26,38)(28,32)(31,40)(35,37);(9,12)(10,16)(11,21)(13,26)(14,24)(15,28)(17,35)(18,32)(19,36)(20,37)(22,39)(23,27)(25,38)(29,30)(31,33)(34,40)Polycyclic presentation:
2^10|b^2=eg,c^2=ij,b^a=be,c^a=cij,c^b=cf,d^b=dij,d^c=di,e^b=eh,e^c=ei,f^a=fi,f^b=fh,f^c=fj,f^d=fj,f^e=fj,g^a=gh,g^b=gh,g^c=ghi,g^d=gj,g^f=gj,i^b=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72341315, #72341347, #72341443, … |
| maximal normal subgroup of | #72341315, #72341347, #72341443, … |
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, 2, F.5*F.7);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);