#141926

order	512
nilpotency class	4
isoclinism	isoclinic class of #141926
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1993
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

301 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cgh,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49122);