#6447131

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2087 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)(9,10,15,21,16,22,14,20)(11,23,26,36,27,40,33,17)(12,28,13,19,31,18,30,38)(24,39,32,34,29,35,25,37);(1,3,5,4)(2,6,7,8)(9,11,16,27)(10,17,22,36)(12,29,31,24)(13,32,30,25)(14,26,15,33)(18,37,28,34)(19,39,38,35)(20,23,21,40);(1,4,5,3)(2,6,7,8)(9,12,14,30,16,31,15,13)(10,18,20,19,22,28,21,38)(11,24,33,25,27,29,26,32)(17,34,40,39,36,37,23,35);(9,13,15,31,16,30,14,12)(10,19,21,18,22,38,20,28)(11,25,26,24,27,32,33,29)(17,35,23,37,36,39,40,34);(9,14,16,15)(10,20,22,21)(11,26,27,33)(12,30,31,13)(17,23,36,40)(18,19,28,38)(24,32,29,25)(34,35,37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354327);