Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C23×#483 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
10282 conjugacy classes of subgroups; only the 4002 normal subgroups are shown.
This group has 10282 conjugacy classes of subgroups — too many to draw. Only its 4002 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4);(5,6);(7,8)(9,12)(10,11)(13,14)(15,16,18,21)(17,19,22,20);(8,11)(12,14);(7,9,10,13)(8,12,11,14)(16,19)(17,22)(20,21)152 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#3114, C23×#3306, C23×#3335, … |
| maximal normal subgroup of | C23×#3114, C23×#3306, C23×#3335, … |
| maximal quotient of | C23×#1853, C22×#26128, C22×#32982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482246);