Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#7430 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(4,7,8,9)(11,12)(13,20,21,16)(14,18,23,17)(15,19)(22,25,26,24);(3,4)(5,9)(6,8)(7,10)(11,13,14,21,15,22,23,26)(12,16,18,20,19,24,17,25)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57466);