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×#29407 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,6,8)(5,9,10,7)(19,20)(21,31)(22,26)(23,30)(24,29)(25,28)(27,33)(32,34);(3,5,6,10)(4,7,8,9)(11,12)(13,16)(14,15)(17,18)(19,21)(20,26)(22,24)(23,33)(25,32)(27,28)(29,31)(30,34);(12,15)(16,18)(19,22,24,21)(20,27,29,34)(23,33,25,32)(26,30,31,28);(4,8)(7,9)(11,13,14,17)(12,16,15,18)(19,23,24,25)(20,28,29,30)(21,32,22,33)(26,34,31,27)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368897);