#6587785

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	#36442	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1617 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)(9,10)(11,14)(12,13)(15,16)(19,26)(24,32)(27,28)(30,31);(2,6)(7,8)(18,22)(19,27)(20,29)(23,25)(26,28)(30,31);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,18)(19,24)(20,23)(21,25)(22,29)(26,32)(27,30)(28,31);(9,11,12,15)(10,14,13,16)(18,23)(19,28)(20,29)(22,25)(26,27)(30,31);(1,4,5,3)(2,8,6,7)(17,19,21,26)(18,24,25,32)(20,28,29,27)(22,30,23,31)

Polycyclic presentation:

2^9|c^2=f,d^2=g,e^2=fi,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, 6494981);