Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#33160 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,6)(8,10)(11,12)(13,15)(14,17)(16,18);(3,4,5,6);(7,8,9,10);(12,15)(13,18)(16,17)116 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6630198, C2×D4×#459, C2×#6578248 |
| maximal normal subgroup of | C2×#6630198, C2×D4×#459, C2×#6578248 |
| maximal quotient of | C2×#512372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537177);