Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,13,12)(11,14,16,15)(17,18,24,29)(19,28,32,41)(20,26,35,45)(21,27,23,39)(22,37,31,25)(30,46,47,43)(33,42,36,44)(34,40,38,48);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,22,24,32,23,31)(18,25,39,41,29,37,27,28)(20,30,36,38,35,47,33,34)(26,40,42,43,45,48,44,46);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20)(18,26)(19,30)(21,36)(22,38)(23,33)(24,35)(25,40)(27,44)(28,46)(29,45)(31,34)(32,47)(37,48)(39,42)(41,43);(2,6)(3,8)(11,16)(14,15)(17,21,24,23)(18,27,29,39)(19,31,32,22)(20,33,35,36)(25,41,37,28)(26,42,45,44)(30,38,47,34)(40,46,48,43);(2,6)(5,7)(17,22,21,32,24,31,23,19)(18,28,39,25,29,41,27,37)(20,34,36,30,35,38,33,47)(26,43,42,48,45,46,44,40)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=ghi,d^2=i,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7514648);