Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#32714 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,27)(22,23)(24,26)(25,29)(28,33)(30,34)(31,32);(3,5)(4,7)(6,10)(8,9)(11,13)(12,15)(14,17)(16,18)(19,21,23,29)(20,25,22,27)(24,33,32,30)(26,34,31,28);(3,5)(4,8)(6,10)(7,9)(12,16)(15,18)(19,22,23,20)(21,28,29,34)(24,26,32,31)(25,33,27,30);(5,10)(7,8)(13,17)(15,18)(20,22)(25,27)(26,31)(28,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538303);