#6580837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#25179	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,19)(20,27,32,24)(21,25,30,28)(23,26,31,29);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,25,32,28)(21,27,30,24)(23,26,31,29);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20,22,32)(18,24,19,27)(21,23,30,31)(25,29,28,26);(1,4,5,3)(2,8,6,7)(20,30)(21,32)(23,31)(24,25)(26,29)(27,28);(17,21,22,30)(18,25,19,28)(20,31,32,23)(24,26,27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6488033);