Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31217 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2448 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)(9,10,13,12)(15,16,19,22,18,23,27,21)(17,24,26,29,25,20,28,30);(3,4)(5,6)(7,9,11,13)(8,12,14,10)(15,17)(16,20)(18,25)(19,26)(21,29)(22,30)(23,24)(27,28);(4,6)(7,10,11,12)(8,13,14,9)(16,21)(19,27)(20,29)(22,23)(24,30)(26,28);(8,14)(9,13)(16,22)(17,25)(19,27)(20,29)(21,23)(24,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410977);