#6622289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574031
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6529485);