#7569696

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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	#59676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 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)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(19,26)(22,30)(25,28)(29,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,19)(18,22)(20,28)(21,26)(23,30)(24,32)(25,27)(29,31);(2,6)(7,8)(10,14)(13,15)(18,23)(22,30)(24,31)(29,32)

Polycyclic presentation:

2^9|b^2=g,c^2=fg,d^2=fg,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,h^d=hi

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, 7476892);