Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13463 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1272 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1272 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) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,7,9)(5,10,6,8)(11,12)(13,19)(14,16)(15,18)(17,21)(20,25)(22,24)(23,26);(3,5,7,6)(4,8,9,10)(12,16)(15,22)(17,23)(19,25);(3,6,7,5)(4,8,9,10)(11,13,14,20)(12,17,16,23)(15,21,22,26)(18,19,24,25)62 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#500955, C2×#499282, C2×#500358, … |
| maximal normal subgroup of | C2×#500955, C2×#499282, C2×#500358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406411);