#411482

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=e,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=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, 318678);