Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12825 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1802 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1080 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);(3,4,7,15,10,16,9,13)(5,17,19,11,20,14,8,25)(6,21,23,32,24,34,22,30)(12,27,28,18,29,26,31,33);(3,5,7,19,10,20,9,8)(4,11,13,17,16,25,15,14)(6,18,22,27,24,33,23,26)(12,21,28,32,29,34,31,30);(3,6)(4,12)(5,18)(7,22)(8,26)(9,23)(10,24)(11,21)(13,28)(14,30)(15,31)(16,29)(17,32)(19,27)(20,33)(25,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400986);