#141920

order	512
nilpotency class	3
isoclinism	isoclinic class of #120247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
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×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 254 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,13)(5,7)(6,11)(8,10)(12,15)(14,16)(17,18,23,19,22,27,40,31,24,28,41,32,21,26,39,30)(20,33,36,45,35,25,42,29,37,43,47,34,38,44,48,46);(1,3,10,16,5,12,4,11)(2,6,13,15,7,14,8,9)(17,19,23,26,22,31,40,18,24,32,41,27,21,30,39,28)(20,34,47,44,35,46,48,33,37,45,36,25,38,29,42,43);(2,7)(6,14)(8,13)(9,15)(17,20,22,35,24,37,21,38)(18,25,26,33,28,44,27,43)(19,29,30,45,32,46,31,34)(23,36,40,42,41,47,39,48)

Polycyclic presentation:

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

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, 49116);