Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27623 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 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)(5,10)(6,8)(7,9)(11,12,14,16)(13,15,17,18)(19,20,24,28)(21,30,31,26)(22,32,33,34)(23,29,25,27);(3,5,7,6)(4,8,9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,26)(22,25)(23,33)(24,31)(27,34)(28,30)(29,32);(3,6,7,5)(4,8,9,10)(13,17)(15,18)(19,22,24,33)(20,26,28,30)(21,23,31,25)(27,32,29,34);(4,9)(8,10)(19,23,24,25)(20,27,28,29)(21,22,31,33)(26,32,30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366101);