Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#2205 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×2−1+4 | C23 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1136 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6)(7,15)(8,18)(9,13)(10,20)(11,16)(12,17)(14,19);(5,7,11,10)(6,12,16,18)(8,19,17,9)(13,15,14,20);(5,8,11,17)(6,13,16,14)(7,9,10,19)(12,20,18,15)80 conjugacy classes — view the full character table.
| maximal subgroup of | Q8×#2205 |
| maximal normal subgroup of | Q8×#2205 |
| maximal quotient of | #72268164 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402959);