Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123243 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1598 |
| commutator, Frattini, cosocle | C2×C22⋊C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
291 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,24,30)(19,31,20,34)(21,37,41,46)(22,29,23,28)(25,35,44,32)(26,33,45,36)(27,39,42,40)(38,47,48,43);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19,24,20)(18,25,30,44)(21,38,41,48)(22,33,23,36)(26,42,45,27)(28,43,29,47)(31,46,34,37)(32,39,35,40);(1,4)(2,8,10,9)(3,12)(5,6)(7,11,15,16)(13,14)(17,20,24,19)(18,26,30,45)(21,35,41,32)(22,33,23,36)(25,42,44,27)(28,31,29,34)(37,43,46,47)(38,40,48,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30462);