#6353852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62095	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261048);