#203148

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

754 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)(3,11,13,16,14,8,12,15)(17,18,23,26)(19,21,25,22)(20,28,31,27)(24,29,32,30);(2,7,10,9)(8,15,11,16)(17,19,22,18,23,25,21,26)(20,27,30,24,31,28,29,32);(1,3,5,13,6,14,4,12)(2,8,9,16,10,11,7,15)(17,20,22,30,23,31,21,29)(18,24,25,28,26,32,19,27);(1,4,6,5)(2,9,10,7)(3,12,14,13)(8,16,11,15)(17,21,23,22)(18,25,26,19)(20,29,31,30)(24,28,32,27)

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,e^a=eh,f^a=fi,d^b=di,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, 110344);