Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#2062 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5510 conjugacy classes of subgroups; only the 1034 normal subgroups are shown.
This group has 5510 conjugacy classes of subgroups — too many to draw. Only its 1034 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,9,11)(7,12,8,10)(13,14)(15,17)(16,18)(19,20);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20);(5,8,9,7)(6,10,11,12)(17,20)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#24112, C22×#23595, C22×#24350 |
| maximal normal subgroup of | C22×#24112, C22×#23595, C22×#24350 |
| maximal quotient of | C2×#241172 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339193);