Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,28)(22,37)(24,33)(25,32)(27,40)(29,35)(30,36)(31,39)(34,38);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,20,33)(12,27,30,38)(13,24,14,23)(17,34,35,40)(18,32,19,21)(22,28,39,29)(26,31,36,37);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,36)(20,35)(21,34)(23,38)(24,27)(25,39)(32,40)(33,37);(3,8)(5,6)(10,18)(12,28)(13,14)(16,32)(17,35)(19,20)(21,33)(22,38)(23,24)(27,39)(29,30)(34,40)Polycyclic presentation:
2^9|a^2=ef,b^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383555);