#6543324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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	#29667	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1629 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)(17,18,19,25,21,24,27,22)(20,28,30,31,29,32,26,23);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,22,24,25)(20,26,29,30)(23,28,31,32);(10,13)(14,16)(17,20)(18,23)(19,26)(21,29)(22,28)(24,31)(25,32)(27,30);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(18,24)(22,25)(23,31)(28,32);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,19,21,27)(18,22,24,25)(20,26,29,30)(23,28,31,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,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, 6450520);