#6441707

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438886
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	#60456	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2071 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)(9,10,15,20,16,21,14,31)(11,22,25,36,26,35,32,17)(12,18,28,38,29,34,24,37)(13,19,27,39,30,33,23,40);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,27)(13,28)(14,25)(15,32)(16,26)(18,19)(20,35)(21,36)(22,31)(23,29)(24,30)(33,34)(37,39)(38,40);(2,6)(5,8)(9,12,15,28,16,29,14,24)(10,18,20,38,21,34,31,37)(11,23,25,13,26,27,32,30)(17,33,22,40,36,19,35,39);(9,13,15,27,16,30,14,23)(10,19,20,39,21,33,31,40)(11,24,25,12,26,28,32,29)(17,34,22,37,36,18,35,38);(9,14,16,15)(10,20,21,31)(11,25,26,32)(12,28,29,24)(13,23,30,27)(17,35,36,22)(18,37,34,38)(19,39,33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,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,e^c=ei,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, 6348903);