#409659

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
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

765 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)(11,16,24,31,23,32,25,17)(12,26,29,18,27,34,28,35)(22,36,38,40,37,33,39,30);(1,3,5,4)(2,6,8,7)(9,11,14,24,13,23,15,25)(10,16,19,31,21,32,20,17)(12,22,28,38,27,37,29,39)(18,30,26,36,35,40,34,33);(1,3,5,4)(2,7,8,6)(9,11,13,23)(10,17)(12,22,27,37)(14,24,15,25)(16,19)(18,33)(20,32)(21,31)(26,30)(28,38,29,39)(34,40)(35,36);(1,4,5,3)(2,6,8,7)(9,12,15,29,13,27,14,28)(10,18,19,26,21,35,20,34)(11,22,25,39,23,37,24,38)(16,30,31,36,32,40,17,33)

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,g^b=gi,g^c=gi

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, 316855);