#128138

order	512
nilpotency class	4
isoclinism	isoclinic class of #128057
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

579 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,c^b=ch,d^b=dh,e^b=egi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35334);