#6357520

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60801	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264716);