Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9337 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1100 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);(3,4,7,11,8,12,6,10)(5,13,15,9,16,17,14,18)(19,20,23,27,24,26,33,21)(22,29,31,34,30,25,32,28);(3,5)(4,9)(6,15)(7,14)(8,16)(10,17)(11,13)(12,18)(19,21,23,26,24,27,33,20)(22,28,32,29,30,34,31,25);(3,5)(4,9)(6,14)(7,15)(8,16)(10,13)(11,17)(12,18)(19,22)(20,25)(21,28)(23,32)(24,30)(26,29)(27,34)(31,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106832);