Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #8534 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C22×#120 | C23 |
| prime radical | #4941255 | C2 |
| LCS | G ▷ C23×D4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×#120 ▷ C24 ▷ C22 ▷ C1 | |
2828 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 2828 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,26)(3,14,17,27)(4,18,21,10)(5,24,20,29)(6,12,7,13)(9,15,28,16)(11,19,30,25)(22,31,23,32);(1,3)(2,9)(4,19,21,20)(5,23,25,22)(6,16)(7,17)(8,15)(10,29,18,30)(11,31,24,32)(12,14)(13,28)(26,27);(1,4,20,17,8,23,5,16)(2,10,30,28,26,31,11,14)(3,6,22,25,15,7,21,19)(9,12,32,24,27,13,18,29)Polycyclic presentation:
2^10|a^2=hi,b^2=e,c^2=deg,d^2=i,b^a=bf,c^a=cfgi,c^b=cd,d^a=dh,d^b=dg,d^c=dgij,e^c=egi,e^d=ej,f^a=fi,f^c=fhj,f^d=fi,g^b=gj,g^c=gj,h^b=hi,h^c=hi58 conjugacy classes — view the full character table.
| maximal subgroup of | #72233377, #72233409, #72233474, … |
| maximal normal subgroup of | #72233377, #72233409, #72233474, … |
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*F.9);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.4*F.5*F.7);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);