#139775

order	512
nilpotency class	4
isoclinism	isoclinic class of #139773
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23902	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46971);