Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C24 |
| direct product | C2×#7163 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #336 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,14,19,16,13,18,22,17,21,15,20,25,24,26,23);(4,7,9,8)(11,13,15,23,16,21,26,19,17,24,14,22,25,12,18,20)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33920);