#6447017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,27)(21,25)(22,28)(24,26)(29,30)(31,32);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,23,18)(20,29,21,31)(22,28,24,26)(25,30,27,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,23,21)(18,25,19,27)(22,30,24,32)(26,29,28,31);(1,4,5,3)(2,8,6,7)(17,21,23,20)(18,25,19,27)(22,32,24,30)(26,29,28,31);(17,22,23,24)(18,26,19,28)(20,30,21,32)(25,29,27,31)

Polycyclic presentation:

2^9|b^2=g,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6354213);