#7157010

order	512
nilpotency class	3
isoclinism	isoclinic class of #7157007
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#44936
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7064206);