#111862

order	512
nilpotency class	3
isoclinism	isoclinic class of #10440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=eh,f^c=fh,g^c=gh

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, 19058);