Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37143 |
| 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 | |
4834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,6,7)(5,9,10,8)(13,17)(15,18)(22,29)(26,31)(28,33)(32,34);(4,7)(8,9)(11,12,14,16)(13,15,17,18)(20,24)(21,27)(26,31)(28,33);(11,13,14,17)(12,15,16,18)(19,20)(21,25)(22,26)(23,24)(27,30)(28,32)(29,31)(33,34);(3,4)(5,8)(6,7)(9,10)(19,21,23,27)(20,25,24,30)(22,28,29,33)(26,32,31,34);(3,5,6,10)(4,8,7,9)(19,22)(20,26)(21,28)(23,29)(24,31)(25,32)(27,33)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541688);