Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20220 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,9)(5,7)(10,12)(11,17)(13,16)(14,18)(19,20,23,26)(21,27,28,34)(22,30,32,25)(24,31,33,29);(3,4,8,12)(5,9,7,10)(6,14,15,17)(11,16,18,13)(21,28)(24,33)(27,34)(29,31);(3,5,8,7)(4,10,12,9)(6,13,15,16)(11,14,18,17)(19,21,23,28)(20,24,26,33)(22,31,32,29)(25,34,30,27);(3,6,8,15)(4,11,12,18)(5,13,7,16)(9,17,10,14)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266769);