Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61435 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6750 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);(4,7)(8,10)(11,12,16,19)(13,17,15,23)(14,21,22,25)(18,24,26,20);(3,4)(5,8)(6,7)(9,10)(11,13,16,15)(12,17,19,23)(14,20,22,24)(18,25,26,21);(11,14)(12,18)(13,20)(15,24)(16,22)(17,25)(19,26)(21,23);(4,7)(8,10)(11,13,16,15)(12,17,19,23)(14,20,22,24)(18,25,26,21);(3,5,6,9)(4,8,7,10)(11,13,16,15)(12,17,19,23)(14,20,22,24)(18,25,26,21)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483968);