Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#23564 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C23 | C2×#120 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3710 conjugacy classes of subgroups; only the 994 normal subgroups are shown.
This group has 3710 conjugacy classes of subgroups — too many to draw. Only its 994 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,7,9)(5,10,6,8)(11,12)(13,16)(14,17)(15,18)(19,23)(20,22)(21,24)(25,26);(3,5,7,6)(4,8,9,10)(11,13)(12,16)(14,19)(15,20)(17,23)(18,22)(21,25)(24,26);(3,6,7,5)(4,8,9,10)(16,22)(23,26);(11,14,15,21)(12,17,18,24)(13,19,20,25)(16,23,22,26)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#418254, C2×#395313, C2×#6445514, … |
| maximal normal subgroup of | C2×#418254, C2×#395313, C2×#6445514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339194);