Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #110030 |
| commutator | C22×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C24×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C24×D4 ▷ C23 ▷ C2 ▷ C1 | |
2335 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)(3,11,13,7)(4,8)(5,18)(6,9)(10,20)(12,28,29,21)(14,22,31,26)(15,23,30,32)(16,25)(17,24)(19,27);(1,3)(2,7)(4,15)(5,14)(6,13)(8,23,24,32)(9,11)(10,26)(12,19)(16,29)(17,30)(18,22)(20,31)(21,25,28,27);(1,4)(2,8)(3,12,13,29)(5,19)(6,17)(7,21,11,28)(9,24)(10,27)(14,15,31,30)(16,20)(18,25)(22,32,26,23)Polycyclic presentation:
2^10|a^2=e,b^2=f,c^2=e,g^2=j,b^a=bd,c^a=cdh,c^b=cdgj,d^a=dj,d^b=di,e^b=ej,f^a=fi,f^c=fh,g^a=gij,g^b=ghi,g^c=gj,g^d=gj,h^a=hj,i^c=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72256008, #72256014, #72256030, … |
| maximal normal subgroup of | #72256008, #72256014, #72256030, … |
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.5);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.4*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);