#6567310

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(4,6)(5,8)(9,10,14,13)(11,16,12,15)(17,18)(19,22)(20,23)(21,25)(24,26)(27,30)(28,31)(29,32);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,23,26,24,20,22,18)(21,27,31,32,29,28,30,25);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,22,26,24,19,23,18)(21,27,30,25,29,28,31,32);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(17,21,24,29)(18,25,26,32)(19,27,20,28)(22,30,23,31);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(21,29)(25,32)(27,28)(30,31)

Polycyclic presentation:

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

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, 6474506);