#112243

order	512
nilpotency class	3
isoclinism	isoclinic class of #112221
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

855 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=eh,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, 19439);