Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,6,4,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,32,30,36,29,35)(27,33,43,45,38,48,39,49)(28,40,44,50,42,34,41,46)(37,51,54,56,53,47,52,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(28,41)(29,31)(33,45)(35,36)(37,52)(39,43)(42,44)(46,50)(48,49)(53,54)(55,56);(1,3,5,4)(2,7,8,6)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27,30,38)(26,33,36,48)(28,37,42,53)(29,39,31,43)(32,45,35,49)(34,47,40,51)(41,52,44,54)(46,55,50,56);(1,3,5,4)(2,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,41,30,42,31,44)(26,34,35,50,36,40,32,46)(27,37,39,52,38,53,43,54)(33,47,49,56,48,51,45,55);(1,4,5,3)(2,7,8,6)(10,16)(11,18)(15,24)(17,19)(27,38)(33,48)(37,53)(39,43)(45,49)(47,51)(52,54)(55,56)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=ghi,d^2=fg,e^2=g,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031123);