#141923

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1985
prime radical	C₂³	#407
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 290 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49119);