Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11956 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #333 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#333 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,11,20,12,30,28,8,13,22,29,6,10,19,26)(5,14,17,27,15,25,21,33,16,31,32,34,18,9,23,24);(3,5,8,16)(4,9)(6,15,20,18)(7,21,22,23)(10,14)(11,27)(12,31)(13,25)(17,19,32,30)(24,28)(26,33)(29,34);(3,6,8,20)(4,10,13,12)(5,15,16,18)(7,19,22,30)(9,14,25,31)(11,26,29,28)(17,21,32,23)(24,27,33,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390296);