#106423

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1149 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13619);