Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#33310 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2962 conjugacy classes of subgroups; only the 1330 normal subgroups are shown.
This group has 2962 conjugacy classes of subgroups — too many to draw. Only its 1330 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(4,6)(7,8,10,12)(9,13,14,11)(16,19)(17,21)(20,22);(7,9,10,14)(8,11,12,13);(15,16,17,19,18,20,21,22);(3,4,5,6)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6631026, #72224775, #72224845, … |
| maximal normal subgroup of | C2×#6631026, #72224775, #72224845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538155);