Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#51529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6814 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)(23,24,29,31)(25,32,26,27)(28,36,37,30)(33,35,34,38);(8,12)(9,15)(10,18)(20,22)(23,25,29,26)(24,27,31,32)(28,33,37,34)(30,35,36,38);(3,4)(5,6)(7,8)(9,16)(10,14)(11,12)(13,15)(17,22)(18,19)(20,21)(23,26,29,25)(24,27,31,32)(28,34,37,33)(30,35,36,38);(7,9)(8,13)(10,17)(11,15)(12,16)(14,20)(18,21)(19,22)(23,27)(24,25)(26,31)(28,35)(29,32)(30,33)(34,36)(37,38);(7,10,11,18)(8,14,12,19)(9,17,15,21)(13,20,16,22)(23,28)(24,30)(25,33)(26,34)(27,35)(29,37)(31,36)(32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614723);