#6544441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451637);