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 | C2 | #251827 |
| commutator, Frattini, cosocle | C23×D4 | C24 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
2747 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 2747 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(1,2)(3,8)(4,9)(5,16)(6,7)(10,17)(11,22)(12,21)(13,18)(14,24)(15,19)(20,26)(23,28)(25,30)(27,29)(31,32);(2,7)(3,11)(4,14)(8,20)(9,23)(10,16)(13,26)(15,28)(18,22)(19,24)(21,29)(30,31);(1,3,5,11)(2,8,10,22)(4,12,14,25)(6,13,17,26)(7,18,16,20)(9,21,24,31)(15,27,28,32)(19,29,23,30);(1,4,6,15)(2,9,7,19)(3,12,13,27)(5,14,17,28)(8,21,18,29)(10,24,16,23)(11,25,26,32)(20,30,22,31)Polycyclic presentation:
2^10|c^2=e,d^2=j,b^a=bj,c^a=cf,c^b=ce,d^a=dfg,d^b=de,e^a=ei,f^b=fij,f^c=fi,f^d=fh,f^e=fj,g^c=ghi,g^d=gh,h^c=hj,i^c=ij,i^d=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72258243, #72314667, #72314685, … |
| maximal normal subgroup of | #72258243, #72314667, #72314685, … |
| maximal quotient of | #72258411 |
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, 3, F.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);