Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483839 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,13,20,15,18,14,31)(11,21,24,17,22,35,25,36)(12,26,27,37,30,16,29,19)(23,34,40,32,28,38,39,33);(2,6)(3,7)(10,16)(11,22)(12,27)(13,14)(17,33)(18,26)(19,31)(20,37)(21,34)(23,39)(28,40)(29,30)(32,36)(35,38);(1,3,5,6,4,7,8,2)(9,11)(10,17,18,36)(12,28)(13,24)(14,25)(15,22)(16,32,26,33)(19,38,37,34)(20,35,31,21)(23,30)(27,39)(29,40);(10,18)(11,22)(16,26)(19,37)(20,31)(23,28)(24,25)(39,40)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320804, #72320825, #72320866, … |
| maximal normal subgroup of | #72320804, #72320825, #72320866, … |
| maximal quotient of | #72343636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391037);