Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1436 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)(11,13)(12,14)(15,16)(17,18,23,27,24,28,22,37)(19,29,31,41,32,40,21,25)(20,33,35,42,36,26,38,43)(30,45,46,39,47,44,34,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,24,32)(18,25,28,41)(20,30,36,47)(21,22,31,23)(26,39,33,48)(27,40,37,29)(34,38,46,35)(42,44,43,45);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,30)(21,34)(22,38)(23,35)(24,36)(25,39)(27,43)(28,33)(29,44)(31,46)(32,47)(37,42)(40,45)(41,48);(3,8)(5,7)(11,16)(13,15)(17,21,23,19,24,31,22,32)(18,25,27,29,28,41,37,40)(20,34,35,30,36,46,38,47)(26,39,43,44,33,48,42,45);(10,14)(13,15)(17,22,24,23)(18,27,28,37)(19,31,32,21)(20,35,36,38)(25,40,41,29)(26,42,33,43)(30,34,47,46)(39,44,48,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6986790);