Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#29693 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(8,9)(12,13)(15,16)(17,23)(18,20)(19,27)(21,25)(22,26)(24,30)(28,29);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(16,20)(21,29)(22,27)(23,30);(4,6)(7,9,11,8)(10,12,14,13)(15,17,18,24)(16,21,20,29)(19,28,26,25)(22,30,27,23);(7,10,11,14)(8,12,9,13)(17,24)(21,29)(23,30)(25,28)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342620);