Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#13674 |
| commutator, Frattini, cosocle | #493 | C24 |
| LCS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1437 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 1437 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 32 points:
(2,7)(3,11,14,18,15,30,13,29)(4,16)(8,24,26,28,22,25,12,27)(9,21)(19,32);(1,2)(3,12,13,22,15,26,14,8)(4,17)(5,21)(6,7)(9,20)(10,19)(11,28,29,24,30,27,18,25)(16,31)(23,32);(2,7)(8,22)(9,21)(10,23)(12,26)(17,31)(24,25)(27,28);(1,3)(2,8)(4,18)(5,13)(6,15)(7,22)(9,12)(10,27)(11,19)(14,20)(16,29)(17,25)(21,26)(23,28)(24,31)(30,32)Polycyclic presentation:
2^10|a^2=gij,b^2=g,f^2=hj,g^2=ij,h^2=j,b^a=bj,c^b=cj,d^a=df,d^b=de,e^a=eg,e^b=ehj,f^a=fj,f^b=fgh,f^d=fh,f^e=fh,g^d=gh,g^e=gij,h^b=hi,h^d=hj,h^e=hj,i^d=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72306556, #72306578, #72338347, … |
| maximal normal subgroup of | #72306556, #72306578, #72338347, … |
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.7*F.9*F.10);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);