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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1386 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)(3,7)(4,6)(5,8)(17,18,24,29)(19,30,32,28)(20,26,35,44)(21,27,23,38)(22,25,36,40)(31,45,47,42)(33,41,37,43)(34,39,46,48);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,36,24,32,23,22)(18,25,38,28,29,40,27,30)(20,31,37,46,35,47,33,34)(26,39,41,42,44,48,43,45);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20,24,35)(18,26,29,44)(19,31,32,47)(21,37,23,33)(22,34,36,46)(25,39,40,48)(27,43,38,41)(28,42,30,45);(2,6)(5,7)(10,14)(13,16)(17,21,24,23)(18,27,29,38)(19,22,32,36)(20,33,35,37)(25,28,40,30)(26,41,44,43)(31,46,47,34)(39,45,48,42);(2,6)(5,7)(17,22,23,32,24,36,21,19)(18,28,27,25,29,30,38,40)(20,34,33,47,35,46,37,31)(26,42,43,39,44,45,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507511);