#6678903

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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	#36455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=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, 6586099);