#6580421

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,23,20,28)(21,24,29,32)(26,30,27,31);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(19,26)(20,27)(21,29)(23,30)(24,32)(28,31);(10,14)(13,16)(17,19,22,20)(18,23,25,28)(21,27,29,26)(24,31,32,30);(1,4,5,3)(2,6,7,8)(19,27)(20,26)(21,29)(23,31)(24,32)(28,30);(17,20,22,19)(18,23,25,28)(21,26,29,27)(24,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,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, 6487617);