#203142

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
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

743 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110338);