Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
906 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,15,21,16,22,18,23)(11,12,17,24,20,14,19,13);(5,6)(7,8)(10,17)(11,23)(12,14)(15,18)(19,22)(20,21);(2,4)(6,8)(9,11,12,21,16,20,14,23)(10,18,19,24,22,15,17,13)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ehi,g^c=ghi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72271414, #72271421, #72271422, … |
| maximal normal subgroup of | #72271414, #72271421, #72271422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55683);