#289213

order	512
nilpotency class	3
isoclinism	isoclinic class of #289187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196409);