Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×#1357 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #331 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×#493 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1426 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 322 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);(5,6,7,12,8,11,15,19,9,13,16,20,17,10,14,18);(6,10)(7,14)(8,17)(11,13)(12,20)(15,16)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#143196, #72215012, #72216332, … |
| maximal normal subgroup of | C2×#143196, #72215012, #72216332, … |
| maximal quotient of | C22×#7107 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390225);