Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3115 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,10)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20)(21,30)(22,26)(23,28)(24,29)(25,33)(27,32)(31,34);(3,5,6,13,7,12,15,16)(4,8,9,11,10,17,14,18)(19,21,24,22)(20,26,29,30)(23,32,25,31)(27,28,34,33);(19,22,24,21)(20,27,29,34)(23,32,25,31)(26,28,30,33);(5,12)(8,17)(11,18)(13,16)(19,23,24,25)(20,28,29,33)(21,31,22,32)(26,27,30,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261760);