#414514

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

791 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,28)(22,32)(23,37)(25,31)(27,34)(29,33)(30,39)(35,38)(36,40);(1,3,5,4)(2,6,7,8)(9,11,15,25)(10,16,20,32)(12,23,29,39)(13,22,14,24)(17,31,19,21)(18,30,34,37)(26,40,33,35)(27,38,28,36);(1,4,5,3)(2,6,7,8)(10,17)(11,22)(13,14)(18,26)(19,20)(21,31)(23,36)(24,25)(27,28)(33,34)(35,40)(38,39);(9,12,15,29)(10,18)(11,23,25,39)(13,28,14,27)(16,30)(17,26)(19,33)(20,34)(21,35)(22,36,24,38)(31,40)(32,37)

Polycyclic presentation:

2^9|b^2=efi,c^2=ef,d^2=gh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=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, 321710);