Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#12856 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2954 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 2954 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16,18,15)(14,17);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(12,15)(13,18)(16,17)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232791, #72232861, #72232994, … |
| maximal normal subgroup of | #72232791, #72232861, #72232994, … |
| maximal quotient of | #72232783, #72232849, #72261675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401057);