#479773

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

606 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)(9,10,13,19,14,15,29,18)(11,20,21,30,23,32,24,16)(12,17,27,34,28,31,25,35)(22,36,37,40,26,39,38,33);(1,3,4,7)(2,5,6,8)(10,15)(11,21)(12,25)(13,29)(16,32)(17,34)(20,30)(23,24)(27,28)(31,35)(33,40)(37,38);(9,11,13,21,14,23,29,24)(10,16,19,20,15,30,18,32)(12,26,25,37,28,22,27,38)(17,33,35,39,31,40,34,36);(2,6)(5,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,31)(16,33)(18,35)(19,34)(20,36)(21,37)(23,26)(24,38)(25,29)(30,40)(32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386969);