Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395181 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 4 generators on 40 points:
(3,7)(5,8)(9,10)(11,20)(12,15)(13,26,28,32)(14,19)(16,22)(17,27,25,38)(18,29)(21,30,36,37)(23,31)(24,33)(34,35,39,40);(1,2,4,6)(3,8,7,5)(9,11,14,22,15,23,29,33)(10,16,18,20,12,24,19,31)(13,21,27,35,28,36,38,40)(17,30,32,34,25,37,26,39);(1,3,4,7)(2,5,6,8)(9,12,15,10)(11,16,23,24)(13,25,28,17)(14,19,29,18)(20,33,31,22)(21,30,36,37)(26,38,32,27)(34,40,39,35);(9,13,14,27,15,28,29,38)(10,17,18,32,12,25,19,26)(11,21,22,35,23,36,33,40)(16,30,20,34,24,37,31,39)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311016);