#437553

order	512
nilpotency class	3
isoclinism	isoclinic class of #437552
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

948 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:

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

Polycyclic presentation:

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

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, 344749);