#496068

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20,16,21,15,29)(11,22,27,40,28,24,33,38)(12,23,30,32,19,13,18,31)(17,34,35,26,37,25,36,39);(2,5)(6,8)(9,11,23,39,15,27,31,34,16,28,13,26,14,33,32,25)(10,17,12,24,20,36,30,40,21,37,19,22,29,35,18,38);(1,3,4,7)(2,6,5,8)(9,12,13,29,15,30,32,10,16,19,23,20,14,18,31,21)(11,24,26,35,27,40,25,17,28,22,39,36,33,38,34,37);(9,13,14,31,16,23,15,32)(10,18,20,12,21,30,29,19)(11,25,27,39,28,34,33,26)(17,24,35,38,37,22,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403264);