#6605527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#37657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,20)(19,26,27,32)(21,25,22,31)(24,29,28,30);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,27)(18,24,20,28)(21,30,22,29)(25,32,31,26);(2,6)(5,8)(10,14)(11,15)(19,27)(24,28)(26,32)(29,30);(9,11)(10,13)(12,15)(14,16)(17,20,23,18)(19,28,27,24)(21,25,22,31)(26,30,32,29);(11,15)(13,16)(17,21,23,22)(18,25,20,31)(19,29,27,30)(24,26,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512723);