#6599104

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#23864	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6506300);