#203165

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
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

717 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:

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

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,d^b=dhi,f^c=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110361);