#437528

order	512
nilpotency class	3
isoclinism	isoclinic class of #437528
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15602
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

933 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344724);