Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C22×#1409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1334 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 202 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)(7,14,15,10)(8,12,18,11)(9,13)(16,19,17,20);(5,7,9,16)(6,10,13,20)(8,17,18,15)(11,19,12,14)68 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#13674, C22×#13584 |
| maximal normal subgroup of | C22×#13674, C22×#13584 |
| maximal quotient of | #72338397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419792);