#7086483

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
prime radical	#30446	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(11,13)(12,14)(15,16)(17,18,22,31,21,25,20,24)(19,26,27,23,30,28,29,32);(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,30)(22,29)(24,28)(25,32)(26,31);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,21,22)(18,24,25,31)(19,27,30,29)(23,28,32,26);(11,16)(13,15)(17,18,20,24,21,25,22,31)(19,28,29,23,30,26,27,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(23,32)(24,31)(26,28)

Polycyclic presentation:

2^9|a^2=fgi,c^2=hi,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 6993679);