#465718

order	512
nilpotency class	3
isoclinism	isoclinic class of #465718
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372914);