#141297

order	512
nilpotency class	4
isoclinism	isoclinic class of #141286
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#460
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

358 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=ei,d^2=ghi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=cg,d^b=dh,f^b=fh,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, 48493);