Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #116356 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C24×D4 | C23 |
| prime radical | C22×#535 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1 | |
2955 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 2955 conjugacy classes of subgroups — too many to draw. Only its 139 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)(3,11,14,7)(4,15,18,8)(5,9,20,26)(6,10)(12,27,30,21)(13,22)(16,32)(17,25)(19,24)(23,29)(28,31);(1,3,5,13)(2,7,9,22)(4,16,17,30)(6,14,20,31)(8,23,24,21)(10,11,26,28)(12,18,29,19)(15,32,25,27);(1,4)(2,8)(3,12)(5,19)(6,18)(7,21)(9,25)(10,15)(11,27)(13,29)(14,30)(16,31)(17,20)(22,23)(24,26)(28,32)Polycyclic presentation:
2^10|a^2=e,b^2=d,b^a=bef,c^a=cfhj,c^b=cdeg,d^a=dh,d^c=dhi,e^b=ehij,e^c=ehi,e^d=ej,f^a=fhij,f^b=fij,g^a=gj,g^c=gj,h^a=hj,h^b=hj,i^a=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72314736, #72314746, #72314791, … |
| maximal normal subgroup of | #72314736, #72314746, #72314791, … |
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.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);