Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9003 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1618 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);(4,6)(7,8,12,14,11,9,10,13)(15,16,19,23,20,24,18,22)(17,25,27,21,28,29,26,30);(3,4,5,6)(7,9,11,8)(10,13,12,14)(15,17)(16,21)(18,27)(19,26)(20,28)(22,29)(23,25)(24,30);(10,12)(13,14)(15,17)(16,21)(18,26)(19,27)(20,28)(22,25)(23,29)(24,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101671);