#6348010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347957
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60946	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6255206);