Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #1343 |
| commutator | C2×#95 | C2×C8 |
| Frattini, cosocle | #7557 | C22 |
| prime radical | C2×#2656 | C4 |
| LCS | G ▷ C2×#95 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×C4 ◁ #7557 ◁ G | |
| LECS | G ▷ #7557 ▷ C24×C4 ▷ C23 ▷ C22 ▷ C1 | |
1198 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,18,19,14)(12,16,22,15)(13,17)(20,24,21,23)(25,26)(27,33)(28,36)(29,32)(30,34,38,40)(31,37)(35,39);(9,11,13,20)(10,14,17,23)(12,21,22,19)(15,18,16,24)(25,27,29,35)(26,30,32,38)(28,34,37,40)(31,39,36,33)Polycyclic presentation:
2^10|a^2=d,b^2=ij,c^2=f,d^2=e,g^2=i,b^a=bc,c^a=cg,c^b=cf,d^b=dfgi,d^c=dhi,f^a=fj,g^a=gh,g^b=gi,g^c=gij,g^d=gi,h^a=hi,h^b=hij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72267813 |
| maximal normal subgroup of | #72267813 |
| maximal quotient of | #72267825 |
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.4);;
gap> SetPower(coll, 2, F.9*F.10);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
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.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);