#6532601

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500010
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	C₂⁴	C₂⁵
prime radical	#62819	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3178 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)(17,18,21,24)(19,25)(20,23,28,30)(22,27)(26,29)(31,32);(1,2)(3,10)(4,7)(5,8)(6,9)(11,15)(12,14)(13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,3,5,9)(2,6,8,10)(4,12,13,11)(7,15,16,14)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32);(1,4,5,13)(2,7,8,16)(3,11,9,12)(6,14,10,15);(2,8)(6,10)(7,16)(14,15)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,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, 6439797);