Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | #31207 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,15,20,16,21,14,32)(11,22,26,33,27,38,25,40)(12,19,13,30,29,36,28,18)(17,24,34,23,35,39,37,31);(1,2,5,7)(3,6,4,8)(9,11,16,27)(10,17)(12,24,29,39)(13,31,28,23)(14,26,15,25)(18,33)(19,22)(20,37)(21,35)(30,40)(32,34)(36,38);(1,3,5,4)(2,6,7,8)(10,18,21,30)(12,13)(14,15)(17,33,35,40)(19,20,36,32)(22,37,38,34)(23,39)(24,31)(25,26)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=fg,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33637);