Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | |
1231 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)(27,37)(28,42)(29,34)(30,35)(31,36)(32,40)(33,44)(38,46)(39,48)(41,47)(43,50)(45,49)(51,54)(52,56)(53,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,40,31,39,30,41)(26,32,35,37,36,47,34,48)(28,38,43,53,45,52,44,54)(33,46,49,51,50,56,42,55);(11,17)(14,20)(18,23)(21,24)(25,28,29,43,31,45,30,44)(26,33,35,49,36,50,34,42)(27,38,40,53,39,52,41,54)(32,46,37,51,47,56,48,55);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,39)(32,47)(37,48)(38,52)(40,41)(46,56)(51,55)(53,54);(1,4,5,3)(2,6,8,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,36,35)(27,40,39,41)(28,43,45,44)(32,48,47,37)(33,42,50,49)(38,53,52,54)(46,55,56,51)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477660);