#6628073

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(9,10,13,14)(11,15,12,16)(17,18,21,25)(19,26,22,23)(20,28,29,24)(27,31,30,32);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,31,28,32);(9,11,13,12)(10,15,14,16)(18,23)(19,22)(20,29)(24,32)(25,26)(28,31);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(17,20,21,29)(18,24,25,28)(19,27,22,30)(23,31,26,32);(9,12,13,11)(10,16,14,15)(20,29)(24,28)(27,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=hi,d^2=fhi,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, 6535269);