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 | #45094 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,27,21,28)(19,25,34,36,30,40,31,39)(20,26,35,37,33,42,32,41)(29,38,45,46,44,47,43,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(21,23)(25,36)(27,28)(29,43)(31,34)(33,35)(37,41)(39,40)(44,45)(46,48);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,29)(21,31)(22,30)(23,34)(24,36)(26,38)(27,40)(28,39)(32,43)(33,44)(35,45)(37,46)(41,48)(42,47);(3,8)(5,7)(11,16)(13,15)(17,20,23,35,22,33,21,32)(18,26,24,37,27,42,28,41)(19,29,34,45,30,44,31,43)(25,38,36,46,40,47,39,48);(10,14)(13,15)(18,27)(19,30)(20,33)(24,28)(31,34)(32,35)(38,47)(46,48)Polycyclic presentation:
2^9|a^2=fgi,c^2=h,d^2=fgi,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, 6987821);