#6614473

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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)(3,8)(4,7)(5,6)(17,18,23,27,22,19,21,26)(20,24,28,31,30,25,29,32);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,24,30,25)(21,26,23,27)(28,32,29,31);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20,22,30)(18,24,19,25)(21,29,23,28)(26,32,27,31);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,25)(19,24)(21,29)(22,30)(23,28)(26,31)(27,32);(10,13)(11,15)(17,21,22,23)(18,26,19,27)(20,28,30,29)(24,31,25,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=gi,d^2=g,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=dh,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, 6521669);