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×#39312 |
| 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 | |
4916 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);(12,16)(17,18)(19,20)(21,25)(22,29)(23,24)(26,30)(27,33)(28,31)(32,34);(3,4,6,8)(5,9,10,7)(20,24)(25,31)(26,29)(32,33);(3,4)(5,7)(6,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,21,23,28)(20,25,24,31)(22,27,30,34)(26,32,29,33);(11,13,15,14)(12,17,16,18)(19,22,23,30)(20,26,24,29)(21,27,28,34)(25,32,31,33);(3,5,6,10)(4,7,8,9)(11,14,15,13)(12,18,16,17)(19,20)(21,25)(22,26)(23,24)(27,32)(28,31)(29,30)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544210);