Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94431 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C22×D4 | C42 |
| Frattini, cosocle | C2×#390 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#390 ◁ G | |
| LECS | G ▷ C2×#390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5)(6,7)(9,10,12,16,13,15,22,17)(11,18,19,23,21,24,20,14);(1,2)(3,7,8,5)(4,6)(9,11)(10,14,17,18,15,23,16,24)(12,19)(13,21)(20,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=i,e^2=f,g^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=df,g^b=gh,d^c=di,e^c=ehi,g^c=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1687);