Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7969 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,11,8,12,7,10)(5,13,15,9,14,18,16,17)(19,20)(21,29,32,33)(22,26)(23,27)(24,28)(25,31,34,30);(3,5,7,16,8,14,6,15)(4,9,11,18,12,17,10,13)(19,21)(20,25,28,34)(22,30)(23,31)(24,32)(26,33,27,29);(5,14)(9,17)(13,18)(15,16)(19,22,24,23)(20,26,28,27)(21,30,32,31)(25,33,34,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88391);