Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93701 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C23×C4 | C2×C8 |
| Frattini, cosocle | C22⋊#95 | C22 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C22⋊#95 ◁ G | |
| LECS | G ▷ C22⋊#95 ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2)(3,5)(4,7)(6,8)(9,10,12,16)(11,17,18,24,19,14,20,22)(13,15,21,23);(2,5)(3,8)(6,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,22)(18,21)(23,24)Polycyclic presentation:
2^9|a^2=d,c^2=f,d^2=g,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=def,g^b=gi,d^c=dh,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72226647, #72267756 |
| maximal normal subgroup of | #72226647, #72267756 |
| maximal quotient of | #72226645, #72267760, #72267803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1088);