#6571432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgi,c^2=fg,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478628);