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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1630 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)(17,18,24,39,23,29,22,28)(19,30,21,25,33,41,32,27)(20,34,36,44,38,26,37,43)(31,46,35,42,48,45,47,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,25,29,27)(20,31,38,48)(21,24,32,22)(26,40,34,42)(28,41,39,30)(35,36,47,37)(43,45,44,46);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,35)(22,37)(23,38)(24,36)(25,40)(27,42)(28,44)(29,34)(30,45)(32,47)(33,48)(39,43)(41,46);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,32,24,33)(18,27,28,41,29,25,39,30)(20,35,37,31,38,47,36,48)(26,42,44,46,34,40,43,45);(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,21,33,32)(20,36,38,37)(25,41,27,30)(26,43,34,44)(31,47,48,35)(40,45,42,46)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 6987942);