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×#57848 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6008 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)(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,18)(19,20,23,27)(21,26,29,24)(22,25,31,34)(28,33,30,32);(6,10)(8,9)(12,13)(14,18)(19,21,23,29)(20,24,27,26)(22,30,31,28)(25,33,34,32);(4,5)(8,9)(12,13)(16,17)(19,22,23,31)(20,25,27,34)(21,28,29,30)(24,32,26,33);(3,5,7,4)(6,9,10,8)(11,13,15,12)(14,17,18,16)(19,21)(20,26)(22,30)(23,29)(24,27)(25,32)(28,31)(33,34);(3,6,7,10)(4,8,5,9)(11,14,15,18)(12,16,13,17)(19,20)(21,26)(22,25)(23,27)(24,29)(28,33)(30,32)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008965);