#141121

order	512
nilpotency class	4
isoclinism	isoclinic class of #141021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1982
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

495 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ei,d^2=ghi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48317);