Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #213828 |
| commutator | C25 | C23×C4 |
| Frattini, cosocle | C23×D4 | C24 |
| prime radical | #62817 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
3031 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 3031 conjugacy classes of subgroups — too many to draw. Only its 207 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)(12,14)(13,24)(21,25)(22,29)(23,30)(26,27);(1,2,5,10,6,7,17,20)(3,8,13,22,11,18,24,29)(4,9,16,23,15,19,28,30)(12,25,26,32,14,21,27,31);(1,3)(2,8)(4,14)(5,13)(6,11)(7,18)(9,21)(10,22)(12,15)(16,27)(17,24)(19,25)(20,29)(23,31)(26,28)(30,32);(1,4)(2,9)(3,12,11,14)(5,16)(6,15)(7,19)(8,21,18,25)(10,23)(13,27,24,26)(17,28)(20,30)(22,32,29,31)Polycyclic presentation:
2^10|b^2=g,d^2=h,g^2=j,b^a=be,c^a=cj,d^a=di,d^b=df,d^c=de,e^b=ei,e^d=ej,f^b=fh,f^c=fi,g^a=gi,g^d=gh,g^e=gj,h^c=hj,i^b=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72262507, #72262581 |
| maximal normal subgroup of | #72262507, #72262581 |
| maximal quotient of | #72339050 |
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, 2, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);