Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#47338 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4952 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);(3,4,7,5)(6,9,10,8)(11,12,14,16)(13,15,17,18)(19,20)(21,24)(22,26)(23,27)(25,29)(28,32)(30,33)(31,34);(3,4)(5,7)(6,9)(8,10)(19,21,23,29)(20,24,27,25)(22,30,31,28)(26,33,34,32);(4,5)(8,9)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,28)(23,29)(24,27)(26,33)(30,31)(32,34);(3,5,7,4)(6,9,10,8)(19,20)(21,24)(22,26)(23,27)(25,29)(28,32)(30,33)(31,34);(3,6,7,10)(4,8,5,9)(11,13,14,17)(12,15,16,18)(19,22,23,31)(20,26,27,34)(21,28,29,30)(24,32,25,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608505);