Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9068 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #386 |
| prime radical | C24 | #123 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1064 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);(4,8,10,13)(5,11,12,14)(19,20,23,27,24,22,31,26)(21,25,29,33,30,28,34,32);(3,4,6,13,7,10,15,8)(5,9,14,17,12,18,11,16)(19,21,23,29,24,30,31,34)(20,25,27,33,22,28,26,32);(3,5,7,12)(4,9,10,18)(6,14,15,11)(8,16,13,17)(19,22)(20,24)(21,28)(23,26)(25,30)(27,31)(29,32)(33,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102956);