#6620124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,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=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6527320);