#6490764

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1660 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)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,21,25,22,24,20,29)(19,26,28,32,27,31,30,23);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19,22,27)(18,23,24,32)(20,28,21,30)(25,31,29,26);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(18,24)(19,27)(25,29)(28,30);(9,12,14,13)(10,15,11,16)(18,24)(19,27)(25,29)(28,30);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(17,20,22,21)(18,25,24,29)(19,28,27,30)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=gh,d^2=hi,e^2=g,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, 6397960);