#415200

order	512
nilpotency class	3
isoclinism	isoclinic class of #396715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

752 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322396);