#7157006

order	512
nilpotency class	3
isoclinism	isoclinic class of #7157001
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

1783 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:

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

Polycyclic presentation:

2^9|a^2=fi,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=eh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7064202);