Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53235 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5054 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);(5,9)(8,10)(12,15)(13,17)(19,20,23,27)(21,25,29,24)(22,30,32,26)(28,33,31,34);(4,7)(8,10)(12,15)(16,18)(19,21,23,29)(20,24,27,25)(22,31,32,28)(26,34,30,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,34)(30,33)(31,32);(3,5)(4,8)(6,9)(7,10)(11,13,14,17)(12,16,15,18)(19,20)(21,25)(22,26)(23,27)(24,29)(28,34)(30,32)(31,33);(3,4)(5,8)(6,7)(9,10)(19,22,23,32)(20,26,27,30)(21,28,29,31)(24,33,25,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012949);