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 | |
1571 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)(17,18,23,29,24,30,22,28)(19,31,33,27,34,43,21,25)(20,26,36,41,37,42,38,44)(32,45,46,40,47,48,35,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,24,34)(18,25,30,27)(20,32,37,47)(21,22,33,23)(26,39,42,40)(28,31,29,43)(35,38,46,36)(41,48,44,45);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,32)(21,35)(22,38)(23,36)(24,37)(25,39)(27,40)(28,44)(29,41)(30,42)(31,45)(33,46)(34,47)(43,48);(3,8)(5,7)(11,16)(13,15)(17,21,23,19,24,33,22,34)(18,27,29,43,30,25,28,31)(20,35,36,32,37,46,38,47)(26,40,41,48,42,39,44,45);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,33,34,21)(20,36,37,38)(25,31,27,43)(26,41,42,44)(32,35,47,46)(39,48,40,45)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fg,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, 6987940);