Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25488 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(7,8,11,14)(9,13,10,12)(15,16,19,24,20,25,18,23)(17,21,26,22,28,30,27,29);(3,4)(5,6)(7,9,11,10)(8,12,14,13)(15,17,18,27,20,28,19,26)(16,21,23,29,25,30,24,22);(4,6)(15,17,20,28)(16,22,25,29)(18,26,19,27)(21,23,30,24);(7,10,11,9)(8,13,14,12)(15,18,20,19)(16,23,25,24)(17,26,28,27)(21,22,30,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353352);