Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#535 |
| commutator, Frattini, cosocle | C23×C4 | C25 |
| socle | C22 | #34963 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,10,17,11,13,12,28)(7,18,21,34,22,14,23,31)(8,15,25,26,24,27,16,9)(19,29,33,20,35,32,30,36);(2,4)(6,13)(8,24)(16,25)(17,28);(2,4)(5,7,12,23,11,22,10,21)(6,14,28,34,13,18,17,31)(8,19,16,30,24,35,25,33)(9,20,27,29,26,36,15,32);(2,4)(5,8,11,24)(6,15,13,27)(7,19)(9,17,26,28)(10,25,12,16)(14,29)(18,32)(20,31)(21,33)(22,35)(23,30)(34,36);(5,9,12,27,11,26,10,15)(6,16,17,8,13,25,28,24)(7,20,23,29,22,36,21,32)(14,30,31,19,18,33,34,35)Polycyclic presentation:
2^10|a^2=f,c^2=fgh,d^2=hj,e^2=i,f^2=j,i^2=j,b^a=bg,c^a=cgh,c^b=cfghij,d^a=dgj,d^b=dhj,d^c=dfghi,e^a=ei,e^d=eij,g^c=gj,h^c=hj,i^a=ij,i^d=ij112 conjugacy classes — view the full character table.
| maximal subgroup of | #72322003, #72322059, #72322386, … |
| maximal normal subgroup of | #72322003, #72322059, #72322386, … |
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, 3, F.6*F.7*F.8);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, 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.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);