Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#7212 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #336 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
482 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);(3,4,5,8,6,7,9,10)(11,12,15,20,18,19,25,26,17,22,24,23,14,13,16,21);(4,7)(5,9)(11,13,16,23,14,22,24,26,17,19,25,20,18,12,15,21)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34359);