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 | #24033 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | #6442790 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#2932 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
3284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,18,10,19,12,15,14,20,8,22,23,17,13,7)(9,25,27,21,28,16,29,32,30,33,24,26,35,34,31,36);(1,2)(3,4)(5,7,10,22,14,15,23,18)(6,8,17,13,20,11,19,12)(9,26,28,32,30,21,35,36)(16,27,25,29,34,24,33,31);(2,4)(5,8,23,12,14,11,10,13)(6,15,19,22,20,7,17,18)(9,27,35,31,30,24,28,29)(16,26,33,36,34,21,25,32);(5,9,10,28,14,30,23,35)(6,16,19,33,20,34,17,25)(7,21,18,32,15,26,22,36)(8,24,13,31,11,27,12,29)Polycyclic presentation:
2^10|a^2=f,b^2=e,c^2=ej,d^2=i,e^2=j,f^2=i,g^2=ij,h^2=ij,i^2=j,b^a=bg,c^a=cfhi,c^b=cghij,d^a=defhj,d^b=dfh,d^c=dei,e^a=ej,f^b=fi,f^c=fj,g^b=gij,g^c=gj,h^b=hij,h^c=hj,h^d=hj,i^b=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72320542, #72322059, #72322570, … |
| maximal normal subgroup of | #72320542, #72322059, #72322570, … |
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.6);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);