#470596

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

876 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)(3,5)(4,6)(7,8)(9,10,13,18,14,15,25,28)(11,19,20,29,22,16,26,31)(12,17,24,33,23,27,36,38)(21,32,34,39,35,30,37,40);(1,3,4,7)(2,5,6,8)(10,15)(11,20)(13,25)(16,29)(17,27)(19,31)(21,34)(22,26)(24,36)(30,39)(32,40)(35,37);(9,11,14,22)(10,16,15,19)(12,21,23,35)(13,26,25,20)(17,32,27,30)(18,29,28,31)(24,37,36,34)(33,40,38,39);(2,6)(5,8)(9,12)(10,17)(11,21)(13,24)(14,23)(15,27)(16,30)(18,33)(19,32)(20,34)(22,35)(25,36)(26,37)(28,38)(29,39)(31,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,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, 377792);