Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13421 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1316 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 186 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);(4,5)(7,8,12,9)(10,17,19,22)(11,14,20,16)(13,21,15,18);(3,4)(5,6)(7,9,12,8)(10,15,19,13)(11,14,20,16)(17,18,22,21);(3,5,6,4)(7,10)(8,13)(9,15)(11,18)(12,19)(14,17)(16,22)(20,21)62 conjugacy classes — view the full character table.
| maximal subgroup of | #72312273, C2×#500770, C2×#499095, … |
| maximal normal subgroup of | #72312273, C2×#500770, C2×#499095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406290);