Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#386 |
| commutator | C23×C4 | C23×C4 |
| Frattini, cosocle | C24×C4 | C24 |
| prime radical | C26 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
5341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(3,7)(5,8)(9,10)(11,13,15,16)(12,14);(1,2,4,6)(3,5,7,8)(11,15)(13,16)(18,21)(19,23)(22,24);(1,3)(2,5)(4,7)(6,8);(3,7)(5,8)(9,11)(10,13,14,16)(12,15)(17,18,19,21,20,22,23,24)Polycyclic presentation:
2^10|a^2=h,b^2=f,d^2=eghi,g^2=j,c^a=cf,d^a=de,d^b=dg,d^c=df,e^a=ei,e^d=ei,g^b=gj,h^d=hi124 conjugacy classes — view the full character table.
| maximal subgroup of | #72264346, #72264370, #72264395, … |
| maximal normal subgroup of | #72264346, #72264370, #72264395, … |
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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);