#6563274

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
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	#36492	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6470470);