Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(5,6,7,8)(9,10,13,19,15,20,14,16)(11,17,21,31,23,32,28,29)(12,24,25,33,26,34,27,18)(22,36,37,40,38,39,35,30);(10,16)(11,21)(12,25)(13,14)(18,33)(19,20)(23,28)(26,27)(29,31)(30,36)(35,37)(39,40);(1,2,4,3)(9,11,15,23)(10,17,20,32)(12,22,26,38)(13,21,14,28)(16,29,19,31)(18,30,33,40)(24,36,34,39)(25,37,27,35);(1,3,4,2)(9,12)(10,18)(11,22)(13,27)(14,25)(15,26)(16,24)(17,30)(19,34)(20,33)(21,35)(23,38)(28,37)(29,36)(31,39)(32,40)Polycyclic presentation:
2^9|a^2=h,c^2=fi,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292343);