#7157008

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

1751 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=f,e^2=h,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, 7064204);