#6447890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,14,12,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,19,21,27)(18,22,23,31)(20,25,28,26)(24,29,32,30);(9,12,13,11)(10,14,15,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(1,4,5,3)(2,6,7,8)(19,26)(20,28)(22,30)(24,32)(25,27)(29,31);(18,23)(22,31)(24,32)(29,30)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355086);