#7051963

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
prime radical	#59220	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6959159);