Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
At least 21170 conjugacy classes of subgroups; only the 29441 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 7 generators on 24 points:
(3,7)(6,8)(10,14)(12,15)(17,18)(19,24)(20,22)(21,23);(10,14)(11,16)(17,18)(19,22)(20,24)(21,23);(9,10,13,14)(11,15,16,12)(17,19,21,20)(18,22,23,24);(2,5)(3,7)(17,18)(19,22)(20,24)(21,23);(2,5)(6,8)(11,16)(12,15);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,20,21,19)(18,22,23,24);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^10|c^2=h,f^2=h,g^2=ij,c^a=ch,c^b=ci,f^b=fh,f^d=fhij,f^e=fij,g^a=gij,g^d=gj,g^e=gi,g^f=gh166 conjugacy classes — view the full character table.
| maximal subgroup of | #72232927, #72234423, #72260847, … |
| maximal normal subgroup of | #72232927, #72234423, #72260847, … |
| maximal quotient of | #72232274, #72232573, #72300457, … |
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.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);