#148045

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#481
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

726 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=ehi,g^c=gi,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, 55241);