#474186

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

710 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,g^c=gi,h^c=hi,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, 381382);