Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#31052 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,16,20,15,19,26,17)(13,21,22,25,18,14,23,24);(3,5)(4,7)(6,10)(8,9)(11,13)(12,14)(15,18)(16,22)(17,25)(19,21)(20,24)(23,26);(12,17)(13,22)(16,26)(18,23)(19,20)(24,25);(4,8)(5,10)(11,14)(12,18)(13,19)(15,21)(16,25)(17,23)(20,22)(24,26)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410448);