Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C27 | C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
8614 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 8614 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,8,6,7,9,10)(11,12)(13,17,18,15)(14,16);(12,15)(16,17);(4,7)(8,10)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#134033, C2×#150393, #72225868, … |
| maximal normal subgroup of | C2×#134033, C2×#150393, #72225868, … |
| maximal quotient of | #72258104, #72261924, C22×#11499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387438);