Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C211×C4 |
| direct product | C2×C22⋊C42 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C27 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
17882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(11,12,13,14)(16,18);(15,16)(17,18);(4,6)(7,8,9,10)(11,14,13,12);(3,4)(5,6)(7,8)(9,10)200 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#648360, C2×#110086, C2×C22⋊C4×#120, … |
| maximal normal subgroup of | C2×#648360, C2×#110086, C2×C22⋊C4×#120, … |
| maximal quotient of | C2×#356055, C22×C22⋊C42, C22⋊C4×#379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420649);