#6580819

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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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)(18,23)(24,29)(25,32)(26,31);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,25)(20,29)(21,23)(22,31)(24,28)(26,30)(27,32);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19,21,27)(18,24,23,29)(20,30,28,22)(25,26,32,31);(1,4,5,3)(2,8,6,7)(19,20)(22,30)(24,32)(25,29)(26,31)(27,28);(17,20,21,28)(18,25,23,32)(19,22,27,30)(24,31,29,26)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=fi,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, 6488015);