#147954

order	512
nilpotency class	4
isoclinism	isoclinic class of #135932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#442
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

984 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)(3,4)(5,6)(7,8)(10,16,20,32)(11,21)(12,14)(13,29)(17,36,38,40)(18,26,30,19)(22,37,35,39)(23,33)(25,31)(28,34);(2,4)(9,10)(11,22)(12,26)(13,30)(14,32)(15,20)(16,34)(17,24)(18,29)(19,28)(21,36)(23,37)(25,35)(27,38)(31,40)(33,39);(6,8)(9,11,15,25)(10,17,20,38)(12,27,28,24)(13,31,29,21)(14,33,34,23)(16,35,32,22)(18,39,30,37)(19,40,26,36)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,e^c=ei,f^c=fh,h^c=hi

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, 55150);