#379519

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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)(3,7)(4,8)(5,6)(9,10,13,19,15,20,14,17)(11,21,22,16,25,32,24,30)(12,26,27,18,29,33,28,34)(23,37,36,35,39,31,38,40);(2,6)(7,8)(9,11,14,24,15,25,13,22)(10,16,19,32,20,30,17,21)(12,23,28,38,29,39,27,36)(18,31,33,40,34,37,26,35);(1,3,5,4)(2,7,6,8)(10,17)(11,22)(13,14)(16,30)(18,26)(19,20)(23,36)(24,25)(27,28)(31,37)(33,34)(38,39);(1,4,5,3)(2,7,6,8)(9,12,15,29)(10,18)(11,23,25,39)(13,27,14,28)(16,31)(17,26)(19,33)(20,34)(21,35)(22,36,24,38)(30,37)(32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286715);