#203168

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
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	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,3)(4,5)(7,8)(9,10,14,19,15,20,13,18)(11,21,23,32,25,36,24,16)(12,26,30,34,29,17,28,35)(22,38,39,31,27,37,40,33);(1,2,4,3,6,8,5,7)(9,11)(10,16,20,32)(12,27)(13,24)(14,23)(15,25)(17,33,26,31)(18,36,19,21)(22,29)(28,39)(30,40)(34,37,35,38);(1,3,4,8,6,7,5,2)(9,12,13,28,15,29,14,30)(10,17,19,35,20,26,18,34)(11,22,24,40,25,27,23,39)(16,31,21,37,32,33,36,38);(1,4,6,5)(2,3,8,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,29,30)(16,21,32,36)(17,34,26,35)(22,39,27,40)(31,37,33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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,c^b=ch,d^b=dhi,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110364);