Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#30375 |
| 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 | |
1714 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)(13,16)(14,15)(17,18)(19,20)(21,25)(22,30)(23,26)(24,27)(28,32)(29,33)(31,34);(3,4,6,7)(5,8,9,10)(19,21,28,31,23,22,24,29)(20,25,32,34,26,30,27,33);(3,5)(4,8)(6,9)(7,10)(12,15)(16,18)(19,22,23,21)(20,25,26,30)(24,29,28,31)(27,34,32,33);(5,9)(8,10)(11,13,14,17)(12,16,15,18)(20,26)(25,30)(27,32)(33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341113);