#6490694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469944
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1666 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)(9,10,12,15,13,16,11,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(10,14)(11,12)(15,16)(17,19,22,29)(18,23,26,32)(20,28,21,27)(24,31,25,30);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,15,16,14)(17,20,22,21)(18,24,26,25)(19,27,29,28)(23,30,32,31);(9,11,13,12)(10,15,16,14)(17,21,22,20)(18,25,26,24)(19,28,29,27)(23,31,32,30);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(18,26)(23,32)(24,25)(30,31)

Polycyclic presentation:

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

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, 6397890);