#6447888

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,23)(21,30)(22,25)(26,29)(27,31)(28,32);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,15,16)(17,19,22,29)(18,24,25,26)(20,27,30,32)(21,28,23,31);(9,12,13,11)(10,14,15,16)(17,20,22,30)(18,21,25,23)(19,26,29,24)(27,28,32,31);(1,4,5,3)(2,6,7,8)(17,18)(19,27)(20,21)(22,25)(23,30)(24,31)(26,28)(29,32);(17,21,22,23)(18,20,25,30)(19,28,29,31)(24,27,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355084);