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×#57813 |
| 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 | |
5108 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);(3,4,7,9)(5,6,10,8)(12,15)(13,17)(19,20)(21,25)(22,30)(23,24)(26,31)(27,28)(29,34)(32,33);(3,5,7,10)(4,8,9,6)(11,12,14,15)(13,16,17,18)(20,24)(21,27)(22,31)(32,34);(11,12)(13,16)(14,15)(17,18)(19,21,23,27)(20,25,24,28)(22,29,31,33)(26,32,30,34);(3,5)(4,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33);(3,6)(4,5)(7,8)(9,10)(11,13,14,17)(12,16,15,18)(19,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009053);