#7569729

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
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	#59858	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1777 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)(12,14)(13,16)(18,23)(24,30)(25,31)(27,32);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,24)(20,27)(21,23)(22,30)(25,26)(28,31)(29,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,21,22)(18,24,23,30)(20,28,29,26)(25,27,31,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,21,29)(18,25,23,31)(19,26,22,28)(24,32,30,27);(2,6)(7,8)(10,14)(13,15)(17,19,21,22)(18,24,23,30)(20,28,29,26)(25,27,31,32)

Polycyclic presentation:

2^9|b^2=g,c^2=fg,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7476925);