Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#29078 |
| 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 | |
1864 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);(3,4,6,7)(5,9,10,8)(11,12,14,16)(13,17,18,15)(19,20,22,26)(21,25,29,24)(23,27,28,32)(30,34,31,33);(4,7)(8,9)(11,13,14,18)(12,15,16,17)(19,21,28,31,22,29,23,30)(20,24,32,34,26,25,27,33);(12,16)(15,17)(19,21,22,29)(20,25,26,24)(23,31,28,30)(27,33,32,34);(3,5,6,10)(4,8,7,9)(12,16)(15,17)(20,26)(24,25)(27,32)(33,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367771);