#466646

order	512
nilpotency class	3
isoclinism	isoclinic class of #466646
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

914 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)(10,26)(11,12)(15,29)(16,17)(20,21)(24,27)(31,32);(1,2,6,11,7,12,5,8)(3,14,13,23,17,25,16,9)(4,10,18,28,21,26,20,22)(15,27,19,31,29,32,30,24);(1,3,5,13,7,17,6,16)(2,9,8,14,12,23,11,25)(4,19,20,29,21,30,18,15)(10,27,22,31,26,32,28,24);(1,4,5,20,7,21,6,18)(2,10,8,22,12,26,11,28)(3,15,13,19,17,29,16,30)(9,24,14,27,23,31,25,32)

Polycyclic presentation:

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

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, 373842);