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×#20266 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,6)(7,9)(8,10)(11,12)(13,17)(14,21)(15,24)(16,19)(18,20)(22,26)(23,25);(4,9)(5,7)(6,10)(11,13,16,15)(12,17,19,24)(14,22,23,20)(18,21,26,25);(3,5,8,7)(4,6,10,9)(11,14)(12,18)(13,20)(15,22)(16,23)(17,25)(19,26)(21,24);(3,6,8,9)(4,5,10,7)(11,13,16,15)(12,17,19,24)(14,20,23,22)(18,25,26,21)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266818);