#6461884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369080);