#6552245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	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

1275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(10,14)(15,17)(16,21)(22,23)(25,26,29,34,30,33,42,47)(27,31,37,45,36,44,52,55)(28,38,39,46,40,48,41,32)(35,49,51,54,50,56,53,43);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,30,36)(26,31,33,44)(28,35,40,50)(29,37,42,52)(32,43,46,54)(34,45,47,55)(38,49,48,56)(39,51,41,53);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,40)(26,32,33,46)(27,35,36,50)(29,41,42,39)(31,43,44,54)(34,48,47,38)(37,53,52,51)(45,56,55,49);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23)(26,33)(31,44)(32,46)(34,47)(38,48)(43,54)(45,55)(49,56);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(26,33)(27,36)(32,46)(34,47)(35,50)(37,52)(38,48)(51,53)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459441);