#6447135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
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	C₂²×C₄	C₂⁵
prime radical	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2072 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)(3,6)(4,8)(5,7)(9,10,15,20,16,21,14,31)(11,22,25,36,26,35,32,17)(12,18,13,19,29,38,28,40)(23,39,30,37,27,34,24,33);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,27)(13,30)(14,25)(15,32)(16,26)(18,37)(19,39)(20,35)(21,36)(22,31)(23,29)(24,28)(33,38)(34,40);(1,4,5,3)(2,6,7,8)(9,12,14,28,16,29,15,13)(10,18,31,40,21,38,20,19)(11,23,32,24,26,27,25,30)(17,33,35,34,36,37,22,39);(9,13,15,29,16,28,14,12)(10,19,20,38,21,40,31,18)(11,24,25,23,26,30,32,27)(17,34,22,33,36,39,35,37);(9,14,16,15)(10,20,21,31)(11,25,26,32)(12,28,29,13)(17,35,36,22)(18,19,38,40)(23,30,27,24)(33,34,37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fgi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^c=hi,h^d=hi

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, 6354331);