Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #115631 |
| commutator | C23×D4 | C22×C4 |
| Frattini, cosocle | C22×H(5,𝔽2) | C23 |
| prime radical | #6432953 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×H(5,𝔽2) ▷ C23 ▷ C2 ▷ C1 | |
3244 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 3244 conjugacy classes of subgroups — too many to draw. Only its 151 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 — 3 generators on 32 points:
(2,7)(5,18)(8,22)(9,24)(10,21)(11,26)(16,32)(19,23)(20,25)(27,30);(1,2)(3,11)(4,7)(5,9)(6,10)(8,13)(12,26)(14,27)(15,30)(16,20)(17,21)(18,25)(19,31)(22,28)(23,29)(24,32);(1,3)(2,8,10,23)(4,15,17,12)(5,13,18,29)(6,14)(7,19)(9,11)(16,31)(20,30,24,26)(21,22)(25,27)(28,32)Polycyclic presentation:
2^10|c^2=f,b^a=bd,c^a=cg,c^b=ce,d^c=dh,e^a=ehj,e^c=ehj,f^a=fi,f^b=fhj,f^d=fj,f^e=fj,g^c=gi,g^e=gj,h^c=hj,i^b=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | #72265987, #72302078, #72305234, … |
| maximal normal subgroup of | #72265987, #72302078, #72305234, … |
| maximal quotient of | #72265927 |
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.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
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);