#466644

order	512
nilpotency class	3
isoclinism	isoclinic class of #466644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16549
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

900 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 373840);