Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3285 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,4)(9,10,14,19)(11,20,21,33)(12,15,27,31)(13,28,26,18)(16,32,35,22)(17,36,30,24)(23,40,29,37)(25,38,39,34);(5,6)(7,8)(10,15)(11,21)(12,24)(13,26)(16,33)(17,18)(19,31)(20,35)(22,39)(25,32)(27,36)(28,30)(34,37)(38,40);(2,4)(6,8);(9,11,14,21)(10,16,19,35)(12,25,27,39)(13,29,26,23)(15,20,31,33)(17,37,30,40)(18,38,28,34)(22,36,32,24)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,e^d=eh,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408425);