Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C2×#12986 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1869 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1274 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);(3,4,9,14,8,13,27,31,10,15,29,33,7,26,17,18)(5,16,20,34,21,6,23,25,22,24,12,32,19,11,30,28);(3,5,17,30,7,19,29,12,10,22,27,23,8,21,9,20)(4,11,31,34,26,24,14,28,15,6,18,32,13,16,33,25);(3,6)(4,12)(5,18)(7,16)(8,24)(9,28)(10,11)(13,30)(14,21)(15,20)(17,32)(19,33)(22,31)(23,26)(25,29)(27,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401623);