Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31251 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2162 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);(7,8,11,13)(9,10,14,12)(15,16,20,22,19,24,18,23)(17,21,27,28,26,30,25,29);(3,4)(5,6)(7,9,11,14)(8,12,13,10)(15,17,19,26)(16,21,24,30)(18,25,20,27)(22,28,23,29);(4,6)(16,22)(18,20)(21,28)(23,24)(25,27)(29,30);(7,10,11,12)(8,9,13,14)(16,23)(18,20)(21,29)(22,24)(25,27)(28,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411076);