Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153633 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13674 |
| commutator, Frattini, cosocle | #493 | C23 |
| prime radical | #3292 | C22 |
| LCS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 265 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)(3,9)(4,8)(5,10)(6,20)(7,11)(12,21)(13,24)(14,29)(15,25)(16,23)(17,30)(18,22)(19,26)(27,31)(28,32);(2,8,11,23)(4,16)(9,21,25,31)(10,26,20,22)(12,27)(17,28)(18,19)(24,32,29,30);(1,3,7,15)(2,9,11,25)(4,17,16,28)(5,14,6,13)(8,21,23,31)(10,24,20,29)(12,19,27,18)(22,30,26,32)Polycyclic presentation:
2^9|b^2=h,c^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,f^a=fg,g^a=gi,h^a=hi,d^b=di,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 | #72231250, C2×#153634, #72283299, … |
| maximal normal subgroup of | #72231250, C2×#153634, #72283299, … |
| maximal quotient of | C2×#153634, #72309175 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60830);