#6543915

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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,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, 6451111);