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×#57123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5710 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);(3,4,6,9)(5,8,10,7)(11,12)(13,15)(14,17)(16,18)(19,20,24,27)(21,23,26,32)(22,25,31,34)(28,30,29,33);(3,5,6,10)(4,7,9,8)(12,13)(16,17)(21,26)(23,32)(28,29)(30,33);(5,10)(7,8)(11,12,15,13)(14,16,18,17)(19,21,24,26)(20,23,27,32)(22,29,31,28)(25,33,34,30);(3,5)(4,8)(6,10)(7,9)(11,13,15,12)(14,17,18,16)(19,22)(20,25)(21,28)(23,30)(24,31)(26,29)(27,34)(32,33);(3,4)(5,8)(6,9)(7,10)(11,14,15,18)(12,16,13,17)(19,23)(20,26)(21,27)(22,30)(24,32)(25,29)(28,34)(31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007161);