#6625001

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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)(11,15)(12,16)(13,14)(17,18,23,21)(19,27,24,30)(20,29,31,28)(22,25,32,26);(2,6)(7,8)(10,14)(11,12)(17,19,23,24)(18,20,21,31)(22,29,32,28)(25,27,26,30);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,23,31)(18,24,21,19)(22,30,32,27)(25,29,26,28);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(17,21,23,18)(19,20,24,31)(22,25,32,26)(27,29,30,28);(9,12,13,11)(10,16,14,15)(17,22,23,32)(18,25,21,26)(19,28,24,29)(20,30,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6532197);