#115373

order	512
nilpotency class	3
isoclinism	isoclinic class of #9986
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60949	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,d^c=di,e^c=eh,f^c=fh

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, 22569);