Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#17250 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3350 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);(4,7)(8,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,22)(21,24,25,26)(27,28,31,33)(29,32,30,34);(4,7)(5,9)(12,13)(16,17)(20,22)(21,25)(27,29,31,30)(28,32,33,34);(3,4,6,7)(5,8,9,10)(11,13,15,12)(14,16,18,17)(19,21)(20,24)(22,26)(23,25)(27,30,31,29)(28,32,33,34);(3,5)(4,8)(6,9)(7,10)(11,14,15,18)(12,16,13,17)(19,22,23,20)(21,24,25,26)(27,28)(29,34)(30,32)(31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752294);