Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#13050 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2206 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 2206 conjugacy classes of subgroups — too many to draw. Only its 258 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 — 4 generators on 22 points:
(1,2);(4,6)(7,8)(9,15,16,22)(10,18,20,13)(11,14)(12,17)(19,21);(3,4,5,6)(9,16)(12,21)(13,18)(17,19);(7,9,11,16)(8,12,14,21)(10,19,20,17)(13,22,18,15)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#494600, D3×#13050, C2×#500059, … |
| maximal normal subgroup of | C2×#494600, C2×#500059, C2×#500067, … |
| maximal quotient of | C2×#139779, #72236735, #72240576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401790);