Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,32,30,36,29,35)(27,33,44,46,38,49,39,50)(28,40,45,52,43,34,41,47)(37,51,54,56,42,48,53,55);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(26,32)(28,41)(29,31)(33,46)(35,36)(37,53)(39,44)(42,54)(43,45)(47,52)(49,50)(55,56);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,27,30,38)(26,33,36,49)(28,42,43,37)(29,39,31,44)(32,46,35,50)(34,51,40,48)(41,54,45,53)(47,56,52,55);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,28,29,41,30,43,31,45)(26,34,35,52,36,40,32,47)(27,37,39,53,38,42,44,54)(33,48,50,56,49,51,46,55);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(27,38)(33,49)(37,42)(39,44)(46,50)(48,51)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=hi,d^2=fg,e^2=h,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=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105040);