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 ◁ C2×#2932 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,12,9,25)(7,16,21,32,20,13,17,27)(8,14,19,29,22,26,24,31)(18,28,33,36,23,30,34,35);(7,17,20,21)(8,22)(13,27,16,32)(14,26)(18,33,23,34)(19,24)(28,35,30,36)(29,31);(2,4)(6,12)(7,18,20,23)(8,19)(9,10)(13,28,16,30)(14,31)(17,33,21,34)(22,24)(26,29)(27,35,32,36);(5,7,19,34,9,17,8,23,11,20,24,33,10,21,22,18)(6,13,29,36,25,32,14,28,12,16,31,35,15,27,26,30)Polycyclic presentation:
2^10|a^2=i,b^2=h,c^2=h,d^2=ehi,e^2=i,g^2=j,i^2=j,b^a=bh,c^a=cf,d^a=dj,d^b=dg,d^c=de,e^b=ej,e^c=ei,e^d=ej,f^a=fj,g^a=gj,g^c=gj,h^d=hj,i^c=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72250127, #72257330, #72257601, … |
| maximal normal subgroup of | #72250127, #72257330, #72257601, … |
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.9);;
gap> SetPower(coll, 2, F.8);;
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, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
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, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, 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);