#497983

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	C₂×#509	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

569 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)(4,8)(5,7)(9,10)(11,19,22,37)(12,24)(13,16)(14,29)(15,21,31,40)(17,27)(18,28)(20,36,39,33)(23,32,30,38)(25,34)(26,35);(1,3,5,4)(2,6,7,8)(9,11,13,22)(10,15,16,31)(12,20,26,39)(14,30,28,23)(17,32,34,38)(18,36,29,33)(19,24,37,35)(21,27,40,25);(1,4,5,3)(2,6,7,8)(10,16)(15,31)(17,34)(18,29)(19,37)(24,35)(32,38)(33,36);(11,20)(12,25)(14,28)(15,32)(17,35)(18,29)(19,36)(21,30)(22,39)(23,40)(24,34)(26,27)(31,38)(33,37)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,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	#72264744, #72264749, #72264806, …
maximal normal subgroup of	#72264744, #72264749, #72264806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405179);