Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#30300 |
| 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 | |
1930 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)(5,8)(6,7)(9,10)(11,12,14,16)(13,17,18,15)(19,20)(21,28)(22,32)(23,27)(24,26)(25,30)(29,34)(31,33);(4,7)(8,10)(19,21,24,31,23,30,22,29)(20,25,26,34,27,28,32,33);(3,5)(4,8)(6,9)(7,10)(11,13,14,18)(12,15,16,17)(21,29)(22,24)(25,33)(26,32)(28,34)(30,31);(5,9)(8,10)(19,22,23,24)(20,26,27,32)(21,29,30,31)(25,34,28,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341014);