Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#22320 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 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)(5,9)(6,8)(7,10)(11,12)(13,16)(14,15)(17,18)(19,20,24,21,23,26,32,28)(22,25,31,27,30,33,34,29);(3,5)(4,7)(6,10)(8,9)(19,21,24,26,23,28,32,20)(22,29,31,25,30,27,34,33);(5,10)(7,9)(12,15)(16,18)(19,22)(20,25)(21,27)(23,30)(24,31)(26,33)(28,29)(32,34);(4,8)(7,9)(11,13,14,17)(12,16,15,18)(20,26)(21,28)(22,30)(31,34)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283954);