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×#53225 |
| 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);(4,7)(5,9)(13,17)(16,18)(19,20,24,21)(22,27,29,25)(23,28,31,26)(30,32,34,33);(3,4,6,7)(5,8,9,10)(12,15)(16,18)(20,21)(22,29)(23,31)(32,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34);(3,5,6,9)(4,8,7,10)(11,12)(13,16)(14,15)(17,18)(19,23,24,31)(20,26,21,28)(22,30,29,34)(25,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012988);