Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61599 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7910 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)(5,9)(6,8)(7,10)(11,12)(13,21)(14,19)(15,17)(16,20)(18,22)(23,26)(24,25);(3,5)(4,7)(6,10)(8,9)(12,17)(18,25)(19,20)(21,26);(11,13)(12,18)(14,22)(15,23)(16,24)(17,25)(19,26)(20,21);(5,10)(7,9)(11,14,15,16)(12,19,17,20)(13,22,23,24)(18,26,25,21);(4,8)(7,9)(11,14,15,16)(12,20,17,19)(13,22,23,24)(18,21,25,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484403);