#6551944

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,12,13,16)(17,18,23,27,22,20,19,24)(21,28,29,31,30,25,26,32);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,16,14,12)(17,19,22,23)(18,24,20,27)(21,29,30,26)(25,32,28,31);(1,4,5,3)(2,6,7,8)(9,12,15,16)(10,11,14,13)(17,20,22,18)(19,24,23,27)(21,28,30,25)(26,31,29,32);(9,13,15,11)(10,16,14,12)(17,21,22,30)(18,25,20,28)(19,26,23,29)(24,31,27,32);(9,14,15,10)(11,12,13,16)(21,30)(25,28)(26,29)(31,32)

Polycyclic presentation:

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

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, 6459140);