Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13345 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2354 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 2354 conjugacy classes of subgroups — too many to draw. Only its 202 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);(3,4)(5,6)(7,8,12,10)(9,15,18,22)(11,14,20,19)(13,21,16,17);(7,9)(8,13)(10,16)(11,17)(12,18)(14,15)(19,22)(20,21);(4,6)(7,10,12,8)(9,16,18,13)(11,14,20,19)(15,17,22,21)62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#498885, D3×#13345, #72312272, … |
| maximal normal subgroup of | C2×#498885, #72312272, #72312273, … |
| maximal quotient of | #72240610, D3×#13345, C2×#144413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406080);