#6624318

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×#468
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531514);