Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26993 |
| 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 | |
2384 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)(13,20)(14,24)(15,18)(16,19)(17,21)(22,25)(23,26);(3,5,6,10)(4,7,8,9)(11,12,15,24,16,19,14,18)(13,17,22,26,23,25,21,20);(4,8)(7,9)(11,12)(13,17)(14,18)(15,24)(16,19)(20,21)(22,26)(23,25);(4,8)(7,9)(11,13,15,22,16,23,14,21)(12,17,24,26,19,25,18,20)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350916);