#6534839

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1381 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)(3,7)(4,8)(5,6)(17,18,24,21)(19,28,29,26)(20,25,30,31)(22,27,23,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,29)(18,25,21,31)(20,23,30,22)(26,32,28,27);(10,13)(14,16)(17,20,24,30)(18,26,21,28)(19,22,29,23)(25,27,31,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,21,24,18)(19,25,29,31)(20,28,30,26)(22,27,23,32);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,22,24,23)(18,27,21,32)(19,20,29,30)(25,26,31,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6442035);