Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#23730 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C22×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3538 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);(7,8,10,12)(9,13,14,11)(15,16,19,24,20,25,18,23)(17,26,27,22,29,30,28,21);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(15,17)(16,21)(18,27)(19,28)(20,29)(22,25)(23,26)(24,30);(4,6)(15,17)(16,22)(18,27)(19,28)(20,29)(21,25)(23,30)(24,26);(15,18,20,19)(16,23,25,24)(17,27,29,28)(21,26,22,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339602);