Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114856 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60799 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1055 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12,8,14,5,3)(4,13,21,17,22,11,20,16)(6,10,27,26,19,29,24,32)(9,15,23,31,28,30,25,18);(2,10)(3,16)(4,19)(5,23)(6,22)(7,25)(11,15)(12,17)(13,30)(14,29)(18,26)(20,21)(24,27)(31,32);(1,3)(2,7)(4,17)(5,14)(6,26)(8,12)(9,18)(10,24)(11,21)(13,20)(15,23)(16,22)(19,32)(25,30)(27,29)(28,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=eh,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213377, #72213848, #72215744, … |
| maximal normal subgroup of | #72213377, #72213848, #72215744, … |
| maximal quotient of | C2×#114858, #72240281, #72257698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22054);