Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,27)(19,28,31,24)(20,32,33,47)(21,26,23,37)(25,34,41,35)(29,43,44,46)(30,40,36,39)(38,45,42,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,27,28)(20,29,33,44)(21,36,23,30)(25,38,41,42)(26,40,37,39)(32,46,47,43)(34,48,35,45);(10,13)(14,16)(17,20,23,35,22,33,21,34)(18,25,26,32,27,41,37,47)(19,29,30,45,31,44,36,48)(24,38,40,46,28,42,39,43);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,33)(25,41)(29,44)(32,47)(34,35)(38,42)(43,46)(45,48);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,23)(18,26,27,37)(19,30,31,36)(20,34,33,35)(24,39,28,40)(25,32,41,47)(29,45,44,48)(38,43,42,46)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466915);