Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C23×#461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
12706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4);(5,6);(8,10)(11,12,14,17)(13,15,18,16);(7,8,9,10);(12,15)(13,18)(16,17)152 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#3036, C23×#3272, C23×D4×SD16, … |
| maximal normal subgroup of | C23×#3036, C23×#3272, C23×D4×SD16, … |
| maximal quotient of | C22×#24530, C22×#24580 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482165);