#6447035

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2006 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(18,22)(19,23)(20,24)(26,28)(27,29)(30,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32);(1,4,5,3)(2,8,6,7)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32);(19,26)(23,28)(25,31)(30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

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, 6354231);