#6594809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 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)(9,10,12,14)(11,13,15,16)(17,18,22,20)(19,26,28,32)(21,25,23,31)(24,27,30,29);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,20)(24,30)(25,31)(26,32);(11,15)(13,16)(17,19,23,29,22,28,21,27)(18,24,25,26,20,30,31,32);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,18)(19,24,28,30)(21,25,23,31)(26,27,32,29);(11,15)(13,16)(17,21,22,23)(18,25,20,31)(19,27,28,29)(24,26,30,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ehi

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, 6502005);