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 | C24 | #120 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 830 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)(9,10,15,19,16,20,11,21)(12,24,22,14,17,13,18,23);(5,6)(7,8)(9,11,16,15)(10,17,20,12)(13,14,24,23)(18,19,22,21);(2,4)(6,8)(9,12,13,19,16,17,24,21)(10,15,18,14,20,11,22,23)Polycyclic presentation:
2^9|a^2=g,b^2=i,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 | #72271441, #72271450, #72271452, … |
| maximal normal subgroup of | #72271441, #72271450, #72271452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55684);