Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#38731 |
| 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 | |
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);(11,12,15,14)(13,17,18,16)(20,24)(21,27)(26,31)(28,33);(4,7)(8,10)(19,20,23,24)(21,25,27,30)(22,26,29,31)(28,32,33,34);(3,4)(5,8)(6,7)(9,10)(11,13,15,18)(12,16,14,17)(19,20)(21,25)(22,26)(23,24)(27,30)(28,32)(29,31)(33,34);(3,5,6,9)(4,8,7,10)(19,21,23,27)(20,25,24,30)(22,28,29,33)(26,32,31,34);(3,4)(5,8)(6,7)(9,10)(11,14,15,12)(13,17,18,16)(19,22,23,29)(20,26,24,31)(21,28,27,33)(25,32,30,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543027);