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 | |
238 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 238 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:
(2,6)(3,4)(7,8)(9,10,14,21,16,22,15,20)(11,23,25,37,27,39,26,38)(12,28,31,19,13,17,29,33)(18,32,35,24,36,30,34,40);(1,2,5,7)(3,6,4,8)(10,17)(12,29)(13,31)(14,15)(18,23)(19,21)(20,33)(22,28)(24,32)(25,26)(30,40)(34,37)(35,38)(36,39);(1,3,5,4)(2,6,7,8)(9,11)(10,18,22,36)(12,30)(13,32)(14,26)(15,25)(16,27)(17,23,28,39)(19,37,33,38)(20,35,21,34)(24,31)(29,40)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fgi,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=dfi,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, 33645);