#6690765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#35921	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2133 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597961);