#6506996

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500385
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	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 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)(17,18)(19,26)(20,25)(21,23)(22,30)(24,29)(27,32)(28,31);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(18,23)(24,31)(25,30)(26,32);(9,11)(10,13)(12,16)(14,15)(17,19,22,29,21,27,20,28)(18,24,25,26,23,31,30,32);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,27)(24,31)(26,32)(28,29);(10,14)(13,15)(17,20,21,22)(18,25,23,30)(19,28,27,29)(24,26,31,32)

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi

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, 6414192);