Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
549 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)(3,8)(4,6)(5,7)(9,10,14,19,15,21,26,20)(11,17,23,16,24,34,12,25)(13,28,30,37,32,18,31,36)(22,38,27,33,29,35,39,40);(1,2)(3,6)(4,8)(5,7)(9,11,15,24)(10,16)(12,26,23,14)(13,29)(17,19)(18,33,28,40)(20,34)(21,25)(22,32)(27,30)(31,39)(35,36,38,37);(1,3,5,4)(2,6,7,8)(9,12,14,11,15,23,26,24)(10,17,20,25,21,34,19,16)(13,22,31,39,32,29,30,27)(18,33,36,35,28,40,37,38);(1,4,5,3)(2,6,7,8)(9,13,14,31,15,32,26,30)(10,18,20,36,21,28,19,37)(11,22,23,39,24,29,12,27)(16,33,17,35,25,40,34,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,e^d=ei,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, 387020);