#139802

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23925	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 730 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)(3,7,8,5)(4,6)(9,10,14,19)(11,20)(12,25)(13,16,28,18)(15,26,31,22)(17,21)(23,34)(24,36)(27,33)(29,40,30,38)(32,37)(35,39);(1,2)(3,5)(4,6)(7,8)(9,11,21,22,14,24,12,26)(10,15,25,36,19,31,17,20)(13,23,37,38,28,39,27,40)(16,29,33,35,18,30,32,34);(1,3,4,8)(2,5,6,7)(10,16)(13,28)(15,29)(17,32)(18,19)(20,34)(23,39)(25,33)(27,37)(30,31)(35,36)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46998);