#140126

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
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	#435
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 241 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=egi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47322);