Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(4,7)(5,9)(11,12)(13,16)(14,22)(15,20)(17,19)(18,23)(21,25)(24,26);(3,4)(5,8)(6,7)(9,10)(11,13,16,12)(14,21,24,18)(15,20,17,19)(22,23,26,25);(11,14)(12,18)(13,21)(15,23)(16,24)(17,25)(19,22)(20,26);(4,7)(8,10)(11,15,16,17)(12,19,13,20)(14,23,24,25)(18,22,21,26);(3,5,6,9)(4,8,7,10)(12,13)(18,21)(19,20)(22,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486083);