#142182

order	512
nilpotency class	4
isoclinism	isoclinic class of #142125
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1987
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

365 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,d^2=gh,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fhi,h^b=hi

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, 49378);