Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#30491 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,11,17,13,19,12,18)(9,20,22,32,24,31,23,14)(10,15,26,16,27,34,28,35)(21,33,25,36,37,30,38,29);(7,9,13,24)(8,14,19,32)(10,25,27,38)(11,23,12,22)(15,33,34,30)(16,29,35,36)(17,31,18,20)(21,28,37,26);(4,6)(7,10)(8,15)(9,21)(11,26)(12,28)(13,27)(14,29)(16,17)(18,35)(19,34)(20,33)(22,25)(23,38)(24,37)(30,31)(32,36);(4,6)(7,10)(8,16)(9,21)(11,28)(12,26)(13,27)(14,30)(15,17)(18,34)(19,35)(20,36)(22,38)(23,25)(24,37)(29,31)(32,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408098);