#139803

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 682 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)(11,13,21,27)(12,22)(14,28)(15,18)(16,24)(17,30)(19,33,35,40)(20,36,38,26)(23,37,39,34)(25,32)(29,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,18,15,21,27,10)(12,19,23,31,25,35,39,16)(14,20,26,28,30,38,36,17)(22,24,34,33,32,29,37,40);(1,3,4,8)(2,5,6,7)(9,12,15,25)(10,16,18,31)(11,19,21,35)(13,23,27,39)(14,29,30,24)(17,32,28,22)(20,37,38,34)(26,40,36,33)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=hi,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, 46999);