Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60589 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7732 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);(3,4)(5,8)(6,7)(9,10)(11,12,16,18)(13,22,14,20)(15,21,17,25)(19,24,26,23);(4,7)(8,10)(12,18)(19,26)(20,22)(21,25);(11,13,16,14)(12,19,18,26)(15,23,17,24)(20,25,22,21);(3,5,6,9)(4,8,7,10)(11,14,16,13)(12,20,18,22)(15,23,17,24)(19,25,26,21);(11,15,16,17)(12,21,18,25)(13,23,14,24)(19,20,26,22)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482319);