#6486212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
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	#24184	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 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)(3,6)(4,5)(7,8)(9,10,11,15,12,14,16,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(2,5)(6,8)(10,13)(11,16)(14,15)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32);(10,13)(11,16)(14,15)(19,25)(22,29)(26,27)(30,31);(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,24)(22,29)(23,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 6393408);