#409660

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

755 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)(3,7)(4,6)(5,8)(9,10,15,20,13,21,14,19)(11,16,24,32,23,33,25,17)(12,26,30,18,28,36,29,37)(22,38,39,35,27,34,40,31);(1,3,5,4)(2,6,8,7)(9,11,14,24,13,23,15,25)(10,16,19,32,21,33,20,17)(12,27,29,40,28,22,30,39)(18,35,26,34,37,31,36,38);(1,3,5,4)(2,7,8,6)(9,11,13,23)(10,17)(12,22,28,27)(14,24,15,25)(16,19)(18,34)(20,33)(21,32)(26,31)(29,39,30,40)(35,36)(37,38);(1,4,5,3)(2,6,8,7)(9,12,15,30,13,28,14,29)(10,18,19,26,21,37,20,36)(11,22,25,40,23,27,24,39)(16,31,32,38,33,35,17,34)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,g^b=gi,g^c=gi

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, 316856);