#140161

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	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 225 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)(3,10)(4,7)(8,15)(9,13)(12,16)(17,18,22,28,24,27,31,44,23,29,39,26,21,40,20,37)(19,30,34,45,36,42,32,43,35,46,47,48,33,25,38,41);(1,2,7,13,5,9,4,6)(3,8,14,15,10,16,11,12)(17,19,31,32,21,33,22,34,23,35,20,38,24,36,39,47)(18,25,37,48,40,46,26,43,29,42,44,45,27,30,28,41);(1,3,4,11,5,10,7,14)(2,8,6,12,9,16,13,15)(17,20,21,39,23,31,24,22)(18,26,27,37,29,28,40,44)(19,32,33,34,35,38,36,47)(25,41,30,45,42,43,46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 47357);