Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | Q8×#356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C23 | C22×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C2×Q82 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
861 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 861 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5);(1,3,4,8)(2,5,6,7);(9,10,13,17,14,18,11,19)(12,16,20,23,22,24,21,15);(9,11,14,13)(10,15,18,23)(12,21,22,20)(16,17,24,19)65 conjugacy classes — view the full character table.
| maximal subgroup of | Q8×#2205 |
| maximal normal subgroup of | Q8×#2205 |
| maximal quotient of | #72268168 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404796);