#6693474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	#24726	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6600670);