Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×C4×#339 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,6);(7,8,9,10)(11,12)(13,15)(14,17)(16,18);(12,15,17,16)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#353780, #72235495, #72238815, … |
| maximal normal subgroup of | C2×#353780, #72235495, #72238815, … |
| maximal quotient of | C2×#123430, C4×#7867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260324);