Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#30555 |
| 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 | |
2821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,22)(14,24)(15,20)(16,26)(17,21)(18,19)(23,25);(3,5,6,10)(4,7,8,9)(11,13,15,19)(12,18,20,22)(14,21,25,26)(16,23,17,24);(11,14,15,25)(12,19,20,13)(16,22,17,18)(21,23,26,24);(4,8)(5,10)(11,13)(12,14)(15,19)(16,24)(17,23)(18,26)(20,25)(21,22)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408331);