Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C23×#356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×Q8 | C25 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
4386 conjugacy classes of subgroups; only the 1090 normal subgroups are shown.
This group has 4386 conjugacy classes of subgroups — too many to draw. Only its 1090 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 — 5 generators on 22 points:
(1,2);(3,4);(5,6);(7,8,11,15,12,16,9,17)(10,14,18,21,20,22,19,13);(7,9,12,11)(8,13,16,21)(10,19,20,18)(14,15,22,17)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#147778, C2×#147565, C23×#2205 |
| maximal normal subgroup of | C2×#147778, C2×#147565, C23×#2205 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529668);