Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,13,29,14,19,26,18)(9,20,24,38,25,35,36,30)(10,16,12,17,11,27,28,33)(15,23,34,22,32,37,31,21);(3,4)(5,6)(7,9,12,23,13,24,11,22,14,25,28,37,26,36,10,21)(8,15,16,30,18,31,33,35,19,32,27,38,29,34,17,20);(4,6)(8,16)(9,21)(10,12)(11,28)(13,26)(17,18)(19,27)(20,30)(22,25)(23,36)(24,37)(29,33)(31,34)(35,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406507);