#6468119

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#34943
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375315);