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×#30625 |
| 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)(14,23)(15,19)(16,20)(17,24)(18,21)(25,26);(3,5)(4,8)(6,9)(7,10)(11,13,16,15)(12,19,20,22)(14,24,25,21)(17,23,18,26);(3,6,7,8)(4,5,9,10)(11,14,16,25)(12,13,20,15)(17,19,18,22)(21,26,24,23);(4,9)(5,10)(11,15)(12,14)(13,16)(17,23)(18,26)(19,21)(20,25)(22,24)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408535);