Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151142 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | #390 | C23 |
| prime radical | #7734 | C2 |
| LCS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 433 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(14,22)(17,19)(23,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,24)(18,23)(19,22);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,17)(13,18)(15,22)(16,23)(19,21)(20,24)Polycyclic presentation:
2^9|c^2=fh,d^2=f,g^2=i,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=eghi,g^b=gi,d^c=dg,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58424);