#6498485

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473153
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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)(3,6)(4,11)(5,7)(8,12)(9,13)(10,16)(14,15)(17,18,22,24)(19,25,29,26)(20,21,30,31)(23,28,32,27);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,23)(20,26)(21,28)(22,29)(24,32)(25,30)(27,31);(2,7)(3,9)(8,11)(10,15)(17,20,21,24,22,30,31,18)(19,26,28,32,29,25,27,23);(3,9)(6,13)(10,15)(14,16);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)

Polycyclic presentation:

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

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, 6405681);