#128792

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

434 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=i,b^a=bd,d^a=dg,f^a=fgh,d^b=dh,h^b=hi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35988);