Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140281 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #468 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18,16,20,15,19)(11,21,17,13,23,12,22,24);(2,5)(6,7)(9,11,12,18,14,22,24,10,16,23,21,19,15,17,13,20);(2,6)(4,8)(5,7)(9,11,13,10,15,17,21,18,16,23,24,20,14,22,12,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=dh,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=egh,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72342450, #72342453, #72342456, … |
| maximal normal subgroup of | #72342450, #72342453, #72342456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47493);