Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,14,20,19,11,15,8,17,18,7,16,12);(5,7)(6,10)(8,12)(9,19)(11,14)(13,16)(15,18)(17,20);(2,4)Polycyclic presentation:
2^9|a^2=f,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219605, #72224555, #72224621, … |
| maximal normal subgroup of | #72219605, #72224555, #72224621, … |
| maximal quotient of | #72219596, #72287256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36705);