#436733

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=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, 343929);