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×#20164 |
| 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 | |
2023 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);(4,7)(5,9)(8,10)(12,16)(14,23)(17,24)(18,19)(20,25)(21,26);(4,8)(7,10)(11,12,15,19)(13,20,22,17)(14,18,23,16)(21,24,26,25);(3,4)(5,10)(6,8)(7,9)(11,13,15,22)(12,17,19,20)(14,21,23,26)(16,24,18,25);(12,18)(14,23)(16,19)(17,25)(20,24)(21,26)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266710);