Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,6,8)(5,9,10,7)(12,17)(13,20)(16,19)(18,21)(22,24)(25,26);(3,5)(4,7)(6,10)(8,9)(11,12)(13,18)(14,16)(15,17)(19,23)(20,26)(21,24)(22,25);(11,13,15,22)(12,18,17,25)(14,21,23,26)(16,24,19,20);(5,10)(7,9)(11,14)(12,19)(13,21)(15,23)(16,17)(18,20)(22,26)(24,25);(12,17)(14,23)(18,25)(21,26)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483558);