Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10664 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #399 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
740 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)(5,13,14,18,15,17,16,9)(19,20,22,26,23,27,31,24)(21,28,29,33,30,34,32,25);(3,5,6,14,8,15,7,16)(4,9,10,13,12,18,11,17)(20,24)(22,31)(25,28)(26,27)(29,32)(33,34);(3,6,8,7)(4,10,12,11)(5,14,15,16)(9,17,18,13)(19,21,23,30)(20,25,27,33)(22,32,31,29)(24,28,26,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151025);