Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25459 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1980 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);(3,4,6,5)(7,8,11,14)(9,13,10,12)(15,16)(17,23)(18,20)(19,22)(21,25)(24,29)(26,28)(27,30);(3,5,6,4)(7,9,11,10)(8,12,14,13)(16,20)(17,24)(19,26)(21,29)(23,30)(25,27);(4,5)(15,17,18,25)(16,21,20,23)(19,27,26,24)(22,29,28,30);(7,10,11,9)(8,13,14,12)(16,20)(21,23)(22,28)(29,30)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353261);