Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#25750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(4,5)(7,8,13,17)(9,14,20,31)(10,21,23,34)(11,25,12,16)(15,27,33,24)(18,32,35,38)(19,30,26,29)(22,37,36,28);(3,4,6,5)(7,9)(8,14)(10,22)(11,19)(12,26)(13,20)(15,32)(16,29)(17,31)(18,24)(21,37)(23,36)(25,30)(27,35)(28,34)(33,38);(7,10,11,24,13,23,12,27)(8,15,16,21,17,33,25,34)(9,18,19,36,20,35,26,22)(14,28,29,32,31,37,30,38);(3,5,6,4)(10,23)(15,33)(18,35)(21,34)(22,36)(24,27)(28,37)(32,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350503);