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 | 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 | |
1247 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,31,35)(27,36,38,48)(28,40,43,54)(29,34,30,45)(32,44,47,39)(33,41,49,42)(37,51,52,46)(50,55,56,53);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,39,31,38,29,44)(26,32,34,36,35,47,45,48)(28,37,42,53,43,52,41,55)(33,46,40,50,49,51,54,56);(11,17)(14,20)(18,23)(21,24)(25,28,30,42,31,43,29,41)(26,33,34,40,35,49,45,54)(27,37,39,53,38,52,44,55)(32,46,36,50,47,51,48,56);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(32,47)(36,48)(37,52)(39,44)(46,51)(50,56)(53,55);(1,4,5,3)(2,6,8,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,45)(27,39,38,44)(28,41,43,42)(32,48,47,36)(33,40,49,54)(37,53,52,55)(46,56,51,50)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,d^b=di,e^b=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477626);