Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,30,35)(27,36,38,32)(28,41,44,55)(29,34,31,46)(33,43,50,45)(37,52,42,56)(39,51,40,48)(47,54,49,53);(1,3,5,4)(2,6,7,8)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(25,27)(26,32)(28,42)(29,39)(30,38)(31,40)(33,49)(34,48)(35,36)(37,44)(41,52)(43,54)(45,53)(46,51)(47,50)(55,56);(11,18)(14,17)(19,24)(21,23)(25,28,29,43,30,44,31,45)(26,33,46,55,35,50,34,41)(27,37,39,53,38,42,40,54)(32,47,51,52,36,49,48,56);(1,4,5,3)(2,6,7,8)(12,20)(15,22)(19,24)(21,23)(27,38)(32,36)(37,42)(39,40)(47,49)(48,51)(52,56)(53,54);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,30,31)(26,34,35,46)(27,39,38,40)(28,43,44,45)(32,48,36,51)(33,41,50,55)(37,53,42,54)(47,56,49,52)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6467088);