#140157

order	512
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1923
prime radical	C₂³	#433
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

208 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,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=ci,e^c=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47353);