Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125701 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×#95 | C23 |
| prime radical | #2656 | C22 |
| LCS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
728 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(2,5)(3,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,23)(22,24);(1,2,4,6)(3,8,7,5)(18,21)(22,23);(10,13)(11,16)(14,15)(17,19,20,24)(18,22)(21,23)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,d^b=df,e^b=egi,d^c=dg,e^c=ehi,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72267806 |
| maximal normal subgroup of | #72267806 |
| maximal quotient of | #72267773, #72267775, #72267808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32997);