Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1572 conjugacy classes of subgroups — too many to draw. Only its 112 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)(6,9)(7,10)(11,12)(13,14)(15,16,20,19)(17,18);(5,6,10,12,8,11,7,9)(13,15,19,18,17,20,16,14);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284212, #72284450, #72284486, … |
| maximal normal subgroup of | #72284212, #72284450, #72284486, … |
| maximal quotient of | #72291979 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45184);