#203126

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^c=dh,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110322);