Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #30400 |
| commutator, Frattini, cosocle | C23×C8 | C24 |
| prime radical | C2×#33618 | C2 |
| LCS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #20383 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
2057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,15,29,13,27,30,12,17,10,14,9,16,28,32)(7,18,21,36,20,34,31,19,24,26,22,25,23,35,33,8);(1,2)(3,4)(5,6)(7,19)(8,24)(9,13)(10,30)(11,32)(12,17)(14,27)(15,28)(16,29)(18,31)(20,36)(21,34)(22,35)(23,25)(26,33);(2,4)(5,7,9,20,12,24,29,23)(6,8,13,25,17,19,16,36)(10,31,27,22,11,33,28,21)(14,34,32,18,15,35,30,26);(5,8,9,25,12,19,29,36)(6,7,16,23,17,24,13,20)(10,26,27,34,11,18,28,35)(14,22,30,31,15,21,32,33)Polycyclic presentation:
2^10|a^2=h,c^2=e,d^2=i,e^2=j,f^2=ij,g^2=ij,h^2=ij,i^2=j,b^a=bh,c^a=cf,c^b=cfgi,d^b=dghi,d^c=degh,e^a=ej,f^b=fi,f^c=fij,f^d=fj,g^b=gi,g^c=gij,g^d=gj,h^b=hi,h^c=hi,i^b=ij,i^c=ij94 conjugacy classes — view the full character table.
| maximal subgroup of | #72322256, #72322302, #72322543, … |
| maximal normal subgroup of | #72322256, #72322302, #72322543, … |
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, 3, F.5);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
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.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.8);;
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.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);