#6715677

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6622873);