Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140281 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×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 | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 266 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)(4,6,8,7)(10,17,19,11)(12,13)(14,15)(18,20,23,21)(22,24);(2,5)(6,7)(9,10,13,23,15,21,12,11,16,19,22,18,14,20,24,17);(2,6)(4,8)(5,7)(9,11,22,19,15,18,24,20,16,17,13,10,14,23,12,21)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=dh,d^2=g,e^2=h,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 | #72227063, #72227162, #72227176, … |
| maximal normal subgroup of | #72227063, #72227162, #72227176, … |
| maximal quotient of | #72227054, #72227264, #72227265 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47486);