Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1224 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,40)(29,34)(30,46)(31,35)(32,38)(33,44)(37,50)(39,48)(41,47)(42,55)(43,51)(45,49)(52,53)(54,56);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,38,31,39,29,45)(26,32,34,36,35,49,46,48)(28,41,44,56,42,37,43,53)(33,50,40,54,51,47,55,52);(11,17)(14,20)(18,23)(21,24)(25,28,30,44,31,42,29,43)(26,33,34,40,35,51,46,55)(27,37,38,53,39,41,45,56)(32,47,36,52,49,50,48,54);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(28,42)(33,51)(37,41)(40,55)(43,44)(47,50)(52,54)(53,56);(1,4,5,3)(2,6,8,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,46)(27,38,39,45)(28,43,42,44)(32,48,49,36)(33,40,51,55)(37,53,41,56)(47,54,50,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477566);