Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3610 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)(9,13)(10,12)(11,14)(15,16,21,23,20,18,19,29)(17,24,28,30,27,22,26,25);(7,9,10,14)(8,11,12,13)(15,17,19,26,20,27,21,28)(16,22,29,30,18,24,23,25);(4,6)(15,18)(16,20)(17,25)(19,29)(21,23)(22,28)(24,26)(27,30);(8,12)(11,13)(15,19,20,21)(16,23,18,29)(17,26,27,28)(22,25,24,30)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366896);