#119841

order	512
nilpotency class	3
isoclinism	isoclinic class of #119837
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,e^c=eh,f^c=fi,g^c=gi

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, 27037);