#6590199

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
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	#36517	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,22,23)(19,24,27,20)(21,25,29,30)(26,31,32,28);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,23)(21,28)(22,27)(25,26)(29,31)(30,32);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(18,23)(19,20)(24,27)(25,30)(26,28)(31,32);(9,12,13,11)(10,16,15,14)(17,20)(18,19)(21,26)(22,24)(23,27)(25,31)(28,30)(29,32);(1,4,5,3)(2,8,6,7)(17,21,22,29)(18,25,23,30)(19,26,27,32)(20,28,24,31)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497395);