Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#42402 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5228 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);(3,4,6,8)(5,7,9,10)(11,12,15,16)(13,14,17,18)(20,24)(25,27)(26,31)(32,33);(12,16)(13,17)(20,24)(22,30)(25,27)(28,34);(3,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,17)(16,18)(19,20,23,24)(21,27,29,25)(22,26,30,31)(28,33,34,32);(11,14)(12,13)(15,18)(16,17)(19,21,23,29)(20,25,24,27)(22,28,30,34)(26,32,31,33);(3,5,6,9)(4,7,8,10)(19,22,23,30)(20,26,24,31)(21,28,29,34)(25,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548048);