Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11945 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
800 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);(3,4,7,12,8,11,6,10)(5,13,14,18,15,9,16,17)(19,20,23,27,32,25,34,30,24,28,33,26,22,21,29,31);(3,5)(4,9)(6,14)(7,16)(8,15)(10,17)(11,13)(12,18)(19,21)(20,22)(23,27)(24,25)(26,33)(28,32)(29,30)(31,34);(3,6,8,7)(4,10,11,12)(5,14,15,16)(9,17,13,18)(19,22,24,32)(20,25,28,21)(23,29,33,34)(26,31,27,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390266);