Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(17,18,22,27)(19,28,23,24)(20,31,33,39)(21,35,38,26)(25,29,42,34)(30,40,36,43)(32,48,37,47)(41,46,44,45);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,36,38,30)(25,39,42,31)(26,43,35,40)(32,46,37,45)(41,47,44,48);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,34,22,33,19,29)(18,25,28,31,27,42,24,39)(21,37,36,45,38,32,30,46)(26,44,43,48,35,41,40,47);(2,6)(3,8)(11,16)(13,15)(17,21,22,38)(18,26,27,35)(19,30,23,36)(20,32,33,37)(24,40,28,43)(25,41,42,44)(29,45,34,46)(31,47,39,48);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(20,33)(21,38)(25,42)(26,35)(29,34)(30,36)(31,39)(40,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819984);