#6611890

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	#24378	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6519086);