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×#61393 |
| 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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7100 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);(4,8)(9,10)(11,12)(13,20)(14,18)(15,23)(16,19)(17,21)(22,25)(24,26);(3,4,7,8)(5,9,6,10);(3,5,7,6)(4,9,8,10)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(3,6,7,5)(4,9,8,10);(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483890);