#7195471

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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	#59680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31);(2,5)(3,7)(10,13)(11,15)(19,28)(23,31)(27,32)(29,30);(9,10,12,13)(11,14,15,16)(17,19,21,30,22,28,20,29)(18,23,24,27,26,31,25,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,27,31,32);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,25,26,24)(19,29,28,30)(23,32,31,27)

Polycyclic presentation:

2^9|c^2=ghi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi

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, 7102667);