#379517

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

885 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)(11,13)(12,25,29,36)(14,20)(15,21)(16,23)(17,28,37,32)(18,30)(19,33)(22,38,39,34)(24,31)(26,35,40,27);(2,6)(7,8)(9,11,14,23,15,24,33,18)(10,16,19,13,21,30,20,31)(12,26,28,39,29,40,32,22)(17,35,36,38,37,27,25,34);(1,3,5,4)(2,7,6,8)(9,12,15,29)(10,17,21,37)(11,22,24,39)(13,27,31,35)(14,32,33,28)(16,34,30,38)(18,26,23,40)(19,25,20,36);(1,4,5,3)(2,7,6,8)(9,13,14,30,15,31,33,16)(10,18,19,11,21,23,20,24)(12,27,32,38,29,35,28,34)(17,26,25,22,37,40,36,39)

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,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, 286713);