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×#30627 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2183 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)(13,21)(14,19)(15,17)(16,20)(18,25)(22,24)(23,26);(3,5)(4,8)(6,9)(7,10)(11,12,14,19)(13,22,23,18)(15,20,16,17)(21,25,26,24);(3,6,7,8)(4,5,9,10)(12,17)(15,16)(18,26)(19,20)(21,22)(24,25);(4,9)(5,10)(11,13)(12,18)(14,23)(15,25)(16,24)(17,26)(19,22)(20,21)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408539);