Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#1324 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| socle | C23 | C2×#99 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4);(5,6,7,9,8,10,11,12)(13,14)(15,21,23,27)(16,24,25,19)(17,20)(18,22)(26,28);(13,15)(14,18,20,26)(16,22)(17,23)(19,21,24,27)(25,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35532);