Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #9495 |
| commutator | C25 | C2×C42 |
| Frattini, cosocle | C23×#41 | C23 |
| socle | C2 | #110401 |
| prime radical | C22×#525 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1 | |
3062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,6,10,16,22)(3,11,12,24,14,7,18,19)(5,9,15,21,17,23,28,31)(13,20,26,30,27,25,29,32);(1,3,5,13)(2,7,9,20)(4,12,15,26)(6,14,17,27)(8,19,21,30)(10,11,23,25)(16,18,28,29)(22,24,31,32);(7,11)(9,23)(19,24)(21,31)Polycyclic presentation:
2^10|a^2=d,b^2=e,d^2=j,b^a=bg,c^a=cg,c^b=cf,e^a=eh,e^c=ei,f^a=fh,f^b=fi,g^b=gh,g^e=gj,h^b=hj,i^a=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72258419, #72258434, #72258453, … |
| maximal normal subgroup of | #72258419, #72258434, #72258453, … |
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.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);