#470940

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

686 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)(3,6)(4,5)(7,8)(9,10,13,19,15,20,14,17)(11,21,24,31,22,34,25,16)(12,26,27,33,28,18,29,32)(23,35,38,30,36,40,37,39);(2,5)(6,8)(9,11,14,24,15,22,13,25)(10,16,17,21,20,31,19,34)(12,23,27,37,28,36,29,38)(18,30,32,35,26,39,33,40);(1,3,4,7)(2,6,5,8)(10,17,20,19)(11,22)(12,27,28,29)(16,21,31,34)(18,26)(23,37,36,38)(24,25)(32,33);(3,7)(6,8)(9,12)(10,18)(11,23)(13,29)(14,27)(15,28)(16,30)(17,32)(19,33)(20,26)(21,35)(22,36)(24,37)(25,38)(31,39)(34,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^d=ei,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378136);