#7079744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996335
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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:

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

Polycyclic presentation:

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

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, 6986940);