#495950

order	512
nilpotency class	4
isoclinism	isoclinic class of #483873
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 ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 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)(9,10,14,19,15,21,29,20)(11,16,24,32,25,34,39,33)(12,18,26,17,27,35,13,28)(22,31,36,30,37,40,23,38);(2,5)(6,8)(9,11,15,25)(10,16,21,34)(12,22,27,37)(13,23,26,36)(14,24,29,39)(17,30,28,38)(18,31,35,40)(19,32,20,33);(1,3,4,7)(2,6,5,8)(10,17)(12,26)(13,27)(14,29)(16,30)(18,19)(20,35)(21,28)(22,36)(23,37)(24,39)(31,32)(33,40)(34,38);(9,12,14,26,15,27,29,13)(10,18,19,17,21,35,20,28)(11,22,24,36,25,37,39,23)(16,31,32,30,34,40,33,38)

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^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	#72239532, #72239543, #72239558, …
maximal normal subgroup of	#72239532, #72239543, #72239558, …
maximal quotient of	#72320473, C₂×#495950

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403146);