Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25942 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3394 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);(4,6)(7,8,11,14)(9,13,10,12)(15,16)(17,26)(18,28)(19,25)(20,24)(21,23)(22,27)(29,30);(7,9,11,10)(8,12,14,13)(15,17)(16,22)(18,19)(20,27)(21,29)(23,28)(24,26)(25,30);(3,4,5,6)(7,9,11,10)(8,13,14,12)(15,18,19,17,20,29,21,27)(16,22,23,30,24,26,25,28);(7,10,11,9)(8,13,14,12)(15,19,20,21)(16,23,24,25)(17,18,27,29)(22,28,26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354874);