Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30624 |
| 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 | |
2231 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)(5,6)(7,9)(8,10)(11,12)(13,22)(14,24)(15,20)(16,25)(17,21)(18,19)(23,26);(3,5)(4,8)(6,9)(7,10)(11,13)(12,18)(14,25)(15,19)(16,23)(17,24)(20,22)(21,26);(3,6,7,8)(4,5,9,10)(11,14)(12,19)(13,20)(15,26)(16,22)(17,18)(21,23)(24,25);(4,9)(5,10)(11,13,15,19)(12,14,20,26)(16,24,17,23)(18,21,22,25)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408540);