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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 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)(12,20)(15,21)(19,24)(22,23)(25,26,29,34,30,35,42,31)(27,32,38,47,37,46,52,43)(28,33,39,49,41,50,54,44)(36,45,51,56,40,48,53,55);(1,3,4,8)(2,5,6,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(26,31)(28,39)(29,42)(32,43)(34,35)(36,51)(38,52)(40,53)(41,54)(44,49)(46,47)(55,56);(2,6)(5,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27,30,37)(26,32,35,46)(28,40,41,36)(29,38,42,52)(31,43,34,47)(33,48,50,45)(39,53,54,51)(44,56,49,55);(3,8)(5,7)(11,18)(14,17)(19,24)(22,23)(25,28,29,39,30,41,42,54)(26,33,34,49,35,50,31,44)(27,36,38,51,37,40,52,53)(32,45,47,56,46,48,43,55);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(27,37)(32,46)(36,40)(38,52)(43,47)(45,48)(51,53)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=i,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7096175);