Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C25 |
| prime radical | C2×#14715 | C2 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
At least 8213 conjugacy classes of subgroups; only the 1341 normal subgroups are shown.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(10,12);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(17,18,21,23)(19,20,22,24);(2,3)(6,7)(9,10)(11,12)(14,16);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,14,15,16)(17,19)(18,20)(21,22)(23,24);(1,3,4,2)(5,7,8,6)(9,10)(11,12)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^10|b^2=f,d^2=g,e^2=h,b^a=bj,c^a=ci,c^b=ch,d^a=di,d^b=dh,d^c=dg,e^a=ei,e^b=eh,e^c=eh,e^d=ef160 conjugacy classes — view the full character table.
| maximal subgroup of | #72331525, #72331541, #72331639, … |
| maximal normal subgroup of | #72331525, #72331541, #72331639, … |
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.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);