Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,10)(5,9,6,8)(11,12,14,16)(13,17,18,15)(19,20,23,28)(21,29,31,25)(22,26,24,27)(30,33,32,34);(3,5,7,6)(4,8,10,9)(11,13,14,18)(12,15,16,17)(19,21)(20,25)(22,30)(23,31)(24,32)(26,34)(27,33)(28,29);(12,16)(15,17)(19,22)(20,26)(23,24)(27,28)(30,32)(33,34);(3,6,7,5)(4,9,10,8)(12,16)(15,17)(19,22,23,24)(20,27,28,26)(21,30,31,32)(25,33,29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353037);