Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#1601 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
5942 conjugacy classes of subgroups; only the 698 normal subgroups are shown.
This group has 5942 conjugacy classes of subgroups — too many to draw. Only its 698 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) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,8,11)(7,9,12,10)(17,20);(6,9)(7,12)(10,11)(13,14)(15,17)(16,18)(19,20);(13,15)(14,17)(16,19)(18,20)92 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#18132, #72292043, #72292045, … |
| maximal normal subgroup of | C22×#18132, #72292043, #72292045, … |
| maximal quotient of | C22×#12202, C2×#257012, C2×#255092 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253530);