Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3483 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
871 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 871 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,20)(11,21,22,32)(12,26,28,34)(13,19,14,18)(16,25,30,24)(17,29,33,27)(23,37,36,31)(35,39,40,38);(9,11,14,25,15,22,13,24)(10,16,19,32,20,30,18,21)(12,23,29,39,28,36,27,38)(17,31,34,40,33,37,26,35);(1,2,4,6)(3,8,7,5)(11,22)(16,30)(21,32)(23,36)(24,25)(31,37)(35,40)(38,39);(3,7)(5,8)(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,31)(18,26)(19,34)(20,33)(21,35)(22,36)(24,38)(25,39)(30,37)(32,40)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=gi,f^d=fi,g^d=gh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255273);