#6519416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
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	C₂×#2540	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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, 6426612);