#6590748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1566 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=f,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497944);