#6628071

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1827 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,12,16)(17,18)(19,24)(20,29)(21,27)(22,31)(23,26)(25,30)(28,32);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,23,20)(18,24,26,29)(21,30,22,28)(25,31,32,27);(9,11,13,12)(10,15,14,16)(17,20,23,19)(18,25,26,32)(21,30,22,28)(24,31,29,27);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(18,26)(24,29)(25,32)(27,31);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,27,26,31)(19,28,20,30)(24,32,29,25)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=hi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,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, 6535267);