Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1854 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);(3,4,6,8)(5,9,10,7)(11,12,14,15)(13,17,18,16)(19,20)(21,24)(22,26)(23,25)(27,31)(28,33)(29,32)(30,34);(3,5,6,10)(4,7,8,9)(12,15)(16,17)(19,21,22,29,23,28,30,27)(20,24,26,32,25,33,34,31);(12,15)(16,17)(21,27)(22,30)(24,31)(26,34)(28,29)(32,33);(4,8)(7,9)(11,13,14,18)(12,16,15,17)(20,25)(21,28)(26,34)(27,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367688);