#6357815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357798
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	#60968	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1968 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)(3,12)(4,9)(5,10)(6,16)(7,11)(8,13)(14,15)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,5,9,7,14,6,13)(2,8,16,15,11,4,10,12)(18,22)(23,30)(24,29)(25,31);(1,4)(2,9)(3,10)(5,12)(6,15)(7,8)(11,13)(14,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(1,5,7,6)(2,10,11,16)(3,13,14,9)(4,12,8,15)(19,26)(23,30)(25,31)(27,32);(1,5,7,6)(2,10,11,16)(3,9,14,13)(4,15,8,12)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265011);