#463918

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 371114);