Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#29649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,11,14)(9,15,17,12)(10,13,18,20)(16,21,22,19)(23,24)(25,33)(26,29)(27,31)(28,37)(30,35)(32,36)(34,38);(3,4)(5,6)(7,9,11,17)(8,12,14,15)(10,16,18,22)(13,19,20,21)(23,25,27,26)(24,29,31,33)(28,34,36,35)(30,32,38,37);(4,6)(7,10)(8,13)(9,16)(11,18)(12,19)(14,20)(15,21)(17,22)(23,26,27,25)(24,30,31,38)(28,34,36,35)(29,32,33,37);(10,18)(13,20)(16,22)(19,21)(25,26)(29,33)(30,38)(34,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341868);