Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(17,18,22,27,23,28,21,24)(19,29,32,38,30,25,37,40)(20,26,33,43,36,44,35,39)(31,42,46,47,34,41,45,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,33)(21,22)(25,40)(27,28)(29,38)(31,45)(34,46)(35,36)(39,43)(41,42);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,23,30)(18,25,28,29)(20,34,36,31)(21,37,22,32)(24,38,27,40)(26,42,44,41)(33,45,35,46)(39,48,43,47);(3,8)(5,7)(11,16)(13,15)(17,20,22,33,23,36,21,35)(18,26,27,43,28,44,24,39)(19,31,32,46,30,34,37,45)(25,41,40,48,29,42,38,47);(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,32,30,37)(20,35,36,33)(25,40,29,38)(26,39,44,43)(31,46,34,45)(41,48,42,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6989846);