#6620456

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#39148	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 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)(4,6)(5,8)(10,14)(15,16)(17,18)(19,23)(20,26)(21,25)(22,27)(24,28)(29,32)(30,31);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,25)(26,31)(27,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,32)(27,31);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(17,20,22,21)(18,25,27,26)(19,28,29,30)(23,31,32,24);(9,12,13,11)(10,16,14,15)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6527652);