Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #30396 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20384 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(7,15,19,20)(8,21)(12,27,25,16)(13,22)(17,33,23,34)(18,31)(28,36,30,35)(29,32);(1,2,3,4)(5,6,9,14,10,11,24,26)(7,16,20,12,19,27,15,25)(8,22,18,32,21,13,31,29)(17,36,33,28,23,35,34,30);(2,4)(6,11)(7,17,19,23)(8,18)(9,24)(12,28,25,30)(13,32)(15,33,20,34)(16,35,27,36)(21,31)(22,29);(5,7,18,34,24,15,8,23,10,19,31,33,9,20,21,17)(6,12,29,35,26,16,13,28,11,25,32,36,14,27,22,30)Polycyclic presentation:
2^10|a^2=h,b^2=fhi,c^2=h,d^2=ehi,e^2=i,i^2=j,b^a=bf,c^b=cfg,d^a=dghij,d^b=dghi,d^c=de,e^a=ej,e^b=ej,e^c=ei,e^d=ej,f^d=fj,g^b=gj,g^d=gj,h^d=hj,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250127, #72250414, #72257253, … |
| maximal normal subgroup of | #72250127, #72250414, #72257253, … |
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.8);;
gap> SetPower(coll, 2, F.6*F.8*F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.5*F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);