Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(9,10,14,18,15,19,13,30)(11,20,23,36,24,16,22,32)(12,25,29,35,27,34,28,17)(21,33,39,31,38,40,26,37);(1,2,4,5)(3,8,7,6)(9,11,15,24)(10,16,19,20)(12,26,27,39)(13,23,14,22)(17,33,35,40)(18,36,30,32)(21,29,38,28)(25,37,34,31);(9,12)(10,17,19,35)(11,21)(13,28)(14,29)(15,27)(16,31,20,37)(18,25,30,34)(22,26)(23,39)(24,38)(32,40,36,33);(1,3,4,7)(2,6,5,8)(10,18)(13,14)(16,32)(17,25)(19,30)(20,36)(22,23)(26,39)(28,29)(31,40)(33,37)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=efgi,c^2=fh,d^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298362);