#6468121

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1060 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:

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

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,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375317);