#114217

order	512
nilpotency class	3
isoclinism	isoclinic class of #111390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

876 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21413);