#495952

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239529, #72239543, #72239558, …
maximal normal subgroup of	#72239529, #72239543, #72239558, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403148);