Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126351 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10,12,24,14,19,25,17)(11,20,21,15,23,35,36,32)(13,27,26,18,29,16,28,33)(22,34,37,30,39,31,38,40);(1,2,6,3)(4,8,5,7)(9,11,14,23)(10,15)(12,21,25,36)(13,22,29,39)(16,30)(17,20)(18,34)(19,32)(24,35)(26,37,28,38)(27,40)(31,33);(1,3,6,2)(4,8,5,7)(10,16)(12,25)(13,28)(15,30)(17,33)(18,24)(19,27)(20,31)(21,36)(22,38)(26,29)(32,40)(34,35)(37,39)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=fi,d^2=f,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,d^c=dg,e^c=ehi,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, 33704);