Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#8689 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,11,9,6,7,17)(5,12,15,13,16,18,14,10)(19,20)(21,27)(22,24)(23,31)(25,29)(26,30)(28,34)(32,33);(3,5,8,15,9,16,7,14)(4,10,17,18,6,13,11,12)(20,24)(25,33)(26,31)(27,34);(3,6)(4,9)(5,13)(7,17)(8,11)(10,16)(12,15)(14,18)(19,21,22,28)(20,25,24,33)(23,32,30,29)(26,34,31,27)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96536);