Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112156 |
| 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 | |
979 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 979 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:
(2,7,11,24)(3,13)(4,18)(8,25,27,23)(9,22,12,14)(10,21,28,26)(15,17)(16,30)(19,32)(29,31);(1,2)(3,14)(4,8)(5,9)(6,12)(7,17)(10,19)(11,20)(13,24)(15,22)(16,25)(18,28)(21,29)(23,30)(26,31)(27,32);(2,8)(3,15)(4,19)(7,21)(9,10)(11,27)(12,28)(13,17)(14,23)(16,31)(18,32)(22,25)(24,26)(29,30)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213377, #72213902, #72215753, … |
| maximal normal subgroup of | #72213377, #72213902, #72215753, … |
| maximal quotient of | C2×#112156, #72240290, #72240292 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19352);