Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C22×Q8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
24586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2);(3,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12);(13,14,16,18)(15,17,19,20);(14,17)(18,20)200 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#36575, C22×#36428, C22×C22⋊C4×#41, … |
| maximal normal subgroup of | C22×#36575, C22×#36428, C22×C22⋊C4×#41, … |
| maximal quotient of | C22×#13907, C22×C22⋊C4×C4⋊C4, C22×#13882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532537);