Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C2×#35446 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
24530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(4,6)(8,10)(11,12,15,16)(13,17,21,19)(14,18,22,23)(20,24,26,25);(7,8)(9,10)(12,16)(14,22)(17,19)(20,26);(3,4)(5,6)(7,8)(9,10)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26);(11,13)(12,17)(14,20)(15,21)(16,19)(18,24)(22,26)(23,25);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536009);