Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3506 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,8,11)(5,12,15,17)(6,10,7,16)(9,14,18,13)(19,20,24,28)(21,29,30,33)(22,27,23,26)(25,32,34,31);(3,5,8,15)(4,9,11,18)(6,13,7,14)(10,17,16,12)(19,21,24,30)(20,25,28,34)(22,32,23,31)(26,33,27,29);(3,6,8,7)(4,10,11,16)(5,13,15,14)(9,17,18,12)(20,26)(22,23)(25,33)(27,28)(29,34)(31,32);(3,6,8,7)(4,10,11,16)(5,14,15,13)(9,12,18,17)(21,30)(25,34)(29,33)(31,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255375);