#496067

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

559 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)(9,10,15,20,17,21,16,30)(11,22,25,37,26,39,33,38)(12,14,27,32,28,13,19,31)(18,23,34,24,36,40,35,29);(2,5)(6,8)(9,11,14,24,16,25,31,40,17,26,13,29,15,33,32,23)(10,18,12,22,20,35,27,38,21,36,28,39,30,34,19,37);(1,3,4,7)(2,6,5,8)(9,12,14,20,16,27,31,21,17,28,13,30,15,19,32,10)(11,22,24,35,25,38,40,36,26,39,29,34,33,37,23,18);(9,13,15,31,17,14,16,32)(10,19,20,12,21,27,30,28)(11,23,25,24,26,40,33,29)(18,22,34,37,36,39,35,38)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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.

Relationships

maximal subgroup of	#72313255, #72313285, #72313292, …
maximal normal subgroup of	#72313255, #72313285, #72313292, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403263);