Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#31192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2698 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);(3,4,5,6)(7,8,11,14)(9,10,13,12)(15,16,19,21)(17,22,25,28)(18,26,20,23)(24,30,27,29);(7,9,11,13)(8,12,14,10)(16,21)(17,24)(18,20)(22,29)(25,27)(28,30);(4,6)(7,10,11,12)(8,13,14,9)(15,17,19,25)(16,22,21,28)(18,24,20,27)(23,29,26,30);(4,6)(8,14)(9,13)(15,18,19,20)(16,23,21,26)(17,24,25,27)(22,29,28,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410927);