Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#29202 |
| 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 | |
1814 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);(3,4,6,7)(5,9,10,8)(11,12,14,16)(13,15,17,18)(19,20)(21,29)(22,32)(23,27)(24,30)(25,26)(28,34)(31,33);(4,7)(8,9)(11,13,14,17)(12,15,16,18)(19,21)(20,25)(22,30)(23,26)(24,31)(27,29)(28,33)(32,34);(13,17)(15,18)(19,22,24,25,23,33,34,29)(20,26,28,32,27,21,30,31);(3,5,6,10)(4,8,7,9)(20,27)(21,26)(28,30)(31,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370842);