Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | #390 | C23 |
| LCS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 467 conjugacy classes of subgroups — too many to draw. Only its 40 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:
(1,2)(3,5)(4,7)(6,8)(12,20)(14,21)(22,23);(2,5)(3,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24);(9,11,12,18,13,19,20,17)(10,14,15,22,16,23,24,21)Polycyclic presentation:
2^9|c^2=g,d^2=f,g^2=i,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=eghi,h^b=hi,d^c=dg,e^c=eh,h^c=hi,e^d=ehi,g^d=gi,g^e=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72213311 |
| maximal normal subgroup of | #72213311 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58452);