#6532512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3076 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:

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

Polycyclic presentation:

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

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, 6439708);