Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153641 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13674 |
| commutator, Frattini, cosocle | #492 | C23 |
| prime radical | #13681 | C2 |
| LCS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 30 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)(3,12)(4,8)(5,19)(6,10)(7,14)(9,22)(11,15)(13,28)(16,30)(17,29)(18,27)(20,23)(21,24)(25,32)(26,31);(1,3,6,15)(2,7)(4,16,18,13)(5,14,22,21)(8,25)(9,12)(10,24)(11,19)(17,32,31,20)(23,27)(26,30)(28,29);(1,4,6,18)(2,8,10,27)(3,13,15,16)(5,20,22,32)(7,23,24,25)(9,29,19,26)(11,28,12,30)(14,31,21,17)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,g^a=gi,h^a=hi,c^b=ci,e^b=efgh,g^b=gh,d^c=df,f^c=fhi,g^c=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72283007, #72283200, #72283261, … |
| maximal normal subgroup of | #72283007, #72283200, #72283261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60840);