Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57026 |
| 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 | |
5152 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)(5,9)(6,8)(7,10)(12,15)(13,14)(19,20,23,26)(21,27,29,24)(22,25,30,32)(28,33,34,31);(3,4,6,8)(5,7,10,9)(11,12,16,15)(13,18,14,17)(22,30)(25,32)(28,34)(31,33);(11,13,16,14)(12,17,15,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34);(3,4)(5,7)(6,8)(9,10)(11,14,16,13)(12,18,15,17)(19,21,23,29)(20,24,26,27)(22,28,30,34)(25,31,32,33);(3,5)(4,7)(6,10)(8,9)(11,15,16,12)(13,18,14,17)(19,22,23,30)(20,25,26,32)(21,28,29,34)(24,31,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007228);