Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C210 |
| direct product | C22×#2113 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C24 | C22×C8 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
4170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2);(3,4);(5,6)(7,8)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,33)(23,31)(24,32)(28,37)(29,35)(30,36)(34,39)(38,40);(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529622);