Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #35346 |
| commutator, Frattini, cosocle | C25 | C25 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
13381 conjugacy classes of subgroups; only the 601 normal subgroups are shown.
This group has 13381 conjugacy classes of subgroups — too many to draw. Only its 601 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24);(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,23)(22,24);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24);(1,2)(3,5)(4,6)(7,8)(11,18)(12,19)(14,21)(20,24);(1,3)(2,5)(4,7)(6,8)(12,19)(14,21)(15,22)(17,23)Polycyclic presentation:
2^10|d^a=df,d^b=dj,d^c=dh,e^a=efg,e^b=ehj,e^c=ej,f^c=fj,f^e=fi,g^b=gj,g^c=gj,g^d=gi,g^e=gi,h^a=hj85 conjugacy classes — view the full character table.
| Fitting subgroup of | #72227753, #72227778 |
| maximal subgroup of | #72227753, #72227770 |
| maximal normal subgroup of | #72227753, #72227770 |
| maximal quotient of | #72267917 |
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> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);