#6441696

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	#60444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

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,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, 6348892);