Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | |
1229 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)(3,5,7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,28)(20,34)(21,29)(22,40)(23,27)(24,30)(25,32)(26,38)(31,46)(33,43)(35,48)(36,45)(37,44)(39,47)(41,42);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,14,16,15)(17,19,23,32,24,33,21,22)(18,25,29,28,30,40,27,43)(20,35,36,41,38,31,39,37)(26,44,45,46,34,42,47,48);(9,12,13,16)(10,14,11,15)(17,20,24,38)(18,26,30,34)(19,31,33,35)(21,39,23,36)(22,41,32,37)(25,42,40,44)(27,47,29,45)(28,48,43,46);(3,7)(5,8)(10,11)(14,15)(17,21,24,23)(18,27,30,29)(19,22,33,32)(20,36,38,39)(25,43,40,28)(26,45,34,47)(31,37,35,41)(42,48,44,46);(17,22,21,33,24,32,23,19)(18,28,27,25,30,43,29,40)(20,37,39,31,38,41,36,35)(26,46,47,44,34,48,45,42)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=g,d^2=i,e^2=fi,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^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497920);