Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C8 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C26 |
| commutator, socle | C22 | C25×C8 |
| prime radical | C22×#525 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
12233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8);(1,2)(3,5)(4,6)(7,8)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(9,11)(10,15)(12,20)(13,21)(14,19)(16,25)(17,26)(18,27)(22,33)(23,31)(24,32)(28,35)(29,36)(30,37)(34,39)(38,40);(1,3,4,7)(2,5,6,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(2,6)(5,8);(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)Polycyclic presentation:
2^10|b^2=g,d^2=h,g^2=i,i^2=j,d^a=dh,d^c=dj,e^b=eh,f^b=fj304 conjugacy classes — view the full character table.
| maximal subgroup of | #72239383, #72239471, #72308523, … |
| maximal normal subgroup of | #72239383, #72239471, #72308523, … |
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.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);