Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#35309 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 30 points:
(1,2);(4,6)(7,8,10,12)(9,11,13,14)(15,16)(17,21)(18,25)(19,20)(22,26)(23,27)(24,30)(28,29);(7,8)(9,11)(10,12)(13,14)(16,20)(17,23)(22,25)(24,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,21)(18,22)(19,20)(23,27)(24,28)(25,26)(29,30);(7,9,10,13)(8,11,12,14)(15,17,19,23)(16,21,20,27)(18,24,26,29)(22,28,25,30);(15,18,19,26)(16,22,20,25)(17,24,23,29)(21,28,27,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535786);