#7019170

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 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:

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

Polycyclic presentation:

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

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, 6926366);