#6488255

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	#24237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,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, 6395451);