#6586497

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6493693);