#250614

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18222	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,7)(3,5)(4,8)(9,10,15,20)(11,21,25,31)(12,26,29,33)(13,19,14,17)(16,24,32,22)(18,28,34,27)(23,30,38,40)(35,36,39,37);(1,2,6,4)(3,7,5,8)(9,11,15,25)(10,16,20,32)(12,23,29,38)(13,24,14,22)(17,31,19,21)(18,30,34,40)(26,35,33,39)(27,36,28,37);(1,3,6,5)(2,7,4,8)(10,17)(11,22)(13,14)(18,33)(19,20)(21,31)(23,36)(24,25)(26,34)(27,28)(35,39)(37,38);(1,4,6,2)(3,8,5,7)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,30)(17,33)(19,26)(20,34)(21,35)(22,36)(24,37)(25,38)(31,39)(32,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157810);