#497703

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

675 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)(3,5)(4,6)(7,8)(10,15,18,29)(12,23)(13,26)(16,28,32,39)(17,27,30,19)(20,34)(21,37)(24,25)(31,38,40,33)(35,36);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,17)(13,27)(14,18)(15,24)(19,26)(20,31)(21,38)(22,32)(23,29)(25,30)(28,35)(33,37)(34,39)(36,40);(2,6)(5,8)(9,11,14,22)(10,16,18,32)(12,20,25,36)(13,21,26,37)(15,28,29,39)(17,31,30,40)(19,33,27,38)(23,34,24,35);(10,17)(12,24)(13,26)(15,27)(16,31)(18,30)(19,29)(20,35)(21,37)(23,25)(28,38)(32,40)(33,39)(34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264905, #72264923, #72264955, …
maximal normal subgroup of	#72264905, #72264923, #72264955, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404899);