#6625087

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#24218	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fg,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,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, 6532283);