#7019179

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei

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, 6926375);