Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3490 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);(7,8,11,14)(9,13,10,12)(15,16,18,20)(17,23,25,30)(19,27,24,22)(21,29,28,26);(3,4)(5,6)(7,9,11,10)(8,12,14,13)(16,20)(21,28)(22,27)(23,30);(4,6)(7,9,11,10)(8,13,14,12)(15,17,24,29,18,25,19,26)(16,21,27,30,20,28,22,23);(7,10,11,9)(8,13,14,12)(16,20)(17,25)(22,27)(26,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354817);