Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1790 |
| prime radical | #52167 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1168 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(5,6)(7,11,8,13)(9,14)(10,15)(12,16,20,17)(18,19);(1,2)(3,4)(6,11)(7,8)(12,14)(13,15)(16,17)(19,20);(2,4)(5,7)(6,12)(8,10)(9,17)(11,19)(13,14)(15,20)(16,18)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dh,e^c=ei,g^c=gi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250465, #72257236, #72257242, … |
| maximal normal subgroup of | #72250465, #72257236, #72257242, … |
| maximal quotient of | #72265245 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54987);