#112116

order	512
nilpotency class	3
isoclinism	isoclinic class of #112109
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
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

857 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19312);