#6605649

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	#39471	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,32,22,26,21,27)(19,28,29,24,25,31,30,20);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,25)(18,24,26,20)(21,29,23,30)(27,31,32,28);(2,6)(5,8)(10,14)(11,15)(17,20,22,24)(18,25,26,19)(21,31,23,28)(27,29,32,30);(9,11)(10,13)(12,15)(14,16)(18,26)(20,24)(27,32)(28,31);(11,15)(13,16)(17,21,22,23)(18,27,26,32)(19,29,25,30)(20,28,24,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 6512845);