Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153645 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13674 |
| commutator, Frattini, cosocle | #494 | C23 |
| prime radical | #3293 | C22 |
| LCS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ G | |
| LECS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
251 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 251 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)(3,15)(4,9)(5,23)(6,13)(7,10)(8,14)(11,22)(12,17)(16,28)(18,30)(19,29)(20,25)(21,27)(24,31)(26,32);(2,9,14,11)(10,25,13,27)(12,29,23,31)(15,28,30,26);(1,3,8,18)(2,10,14,13)(4,19,22,24)(5,6,17,7)(9,25,11,27)(12,30,23,15)(16,21,32,20)(26,31,28,29)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fg,g^a=gi,h^a=hi,e^b=efgh,g^b=gh,d^c=df,e^c=ei,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 | #72283371, #72283600 |
| maximal normal subgroup of | #72283371, #72283600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60842);