Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47278 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,16)(13,15,17,18)(19,20)(21,25)(22,26)(23,24)(27,29)(28,33)(30,32)(31,34);(3,4,7,10)(5,8,6,9)(20,23)(21,27)(22,30)(33,34);(3,5,7,6)(4,8,10,9)(11,12)(13,15)(14,16)(17,18)(19,21,24,27)(20,25,23,29)(22,31,30,28)(26,34,32,33);(3,4)(5,8)(6,9)(7,10)(11,13,14,17)(12,15,16,18)(19,22,24,30)(20,26,23,32)(21,28,27,31)(25,33,29,34);(3,6,7,5)(4,9,10,8)(19,23,24,20)(21,29,27,25)(22,32,30,26)(28,34,31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608563);