Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(7,8,12,14)(9,15,16,10)(11,19,20,13)(17,18,22,21)(23,24,28,31)(25,32,26,29)(27,35,36,30)(33,37,34,38);(9,16)(10,15)(17,22)(18,21)(23,25,28,26)(24,29,31,32)(27,33,36,34)(30,37,35,38);(3,4)(5,6)(7,9)(8,10)(11,17)(12,16)(13,18)(14,15)(19,21)(20,22)(23,26,28,25)(24,29,31,32)(27,34,36,33)(30,37,35,38);(7,10)(8,9)(11,18)(12,15)(13,17)(14,16)(19,22)(20,21)(23,24)(25,32)(26,29)(27,30)(28,31)(33,38)(34,37)(35,36);(7,11,12,20)(8,13,14,19)(9,17,16,22)(10,18,15,21)(23,27,28,36)(24,30,31,35)(25,33,26,34)(29,37,32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541987);