#6615223

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	#37764	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(3,7)(4,8)(5,6)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(9,10,12,13)(11,14,15,16)(17,19,22,29)(18,23,26,32)(20,28,21,30)(24,31,25,27);(2,6)(7,8)(10,13)(11,15)(17,18)(19,23)(20,25)(21,24)(22,26)(27,28)(29,32)(30,31);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,32,31)

Polycyclic presentation:

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

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, 6522419);