Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7900 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(7,8,12,13)(9,11,14,10)(15,16,19,22,20,23,18,28)(17,24,25,30,27,29,26,21);(4,6)(7,9,11,13,12,14,10,8)(15,17)(16,21,23,30)(18,26)(19,25)(20,27)(22,24,28,29);(7,10,12,11)(8,9,13,14)(15,18,20,19)(16,22,23,28)(17,25,27,26)(21,29,30,24)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88163);