Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2326 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 2326 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 20 points:
(1,2)(3,4)(6,10)(11,16)(12,17)(18,20);(5,6)(7,11)(8,10)(9,12)(13,16)(14,18)(15,17)(19,20);(2,4)(8,15)(13,19);(5,7,9,14)(6,11,12,18)(8,13,15,19)(10,16,17,20)Polycyclic presentation:
2^9|d^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,h^b=hi,e^c=eh,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214565, #72215601, #72232895, … |
| maximal normal subgroup of | #72214565, #72215601, #72232895, … |
| maximal quotient of | #72214368, #72215545, #72231173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406081);