Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C23×#456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
22122 conjugacy classes of subgroups; only the 4002 normal subgroups are shown.
This group has 22122 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) = 128 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4);(5,6);(7,8,12,10)(9,15,18,22)(11,19,21,14)(13,20,16,17);(7,9)(8,13)(10,16)(11,20)(12,18)(14,22)(15,19)(17,21);(7,10,12,8)(9,16,18,13)(11,14,21,19)(15,20,22,17)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72266849, #72266854, C2×#499878, … |
| maximal normal subgroup of | #72266849, #72266854, C2×#499878, … |
| maximal quotient of | C22×#32892 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482056);