#495948

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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

567 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)(11,20,22,37)(12,17)(13,26)(14,24)(15,19)(16,21,31,39)(18,28)(23,34,30,38)(25,36)(27,33,40,32)(29,35);(2,5)(6,8)(9,10)(11,16)(12,24)(13,18)(14,17)(15,19)(20,21)(22,31)(23,32)(25,35)(26,28)(27,34)(29,36)(30,33)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,23)(13,27)(14,30)(15,22)(17,33)(18,34)(19,31)(20,35)(21,25)(24,32)(26,37)(28,39)(29,40)(36,38);(9,12,15,14)(10,17,19,24)(13,28,29,25)(16,31)(18,35,36,26)(21,39)(27,40)(32,33)

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=gi,f^2=h,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	#72343241, #72343260, #72343271
maximal normal subgroup of	#72343241, #72343260, #72343271
maximal quotient of	#72343695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403144);