#139804

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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#537 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 669 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,e^2=i,f^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, 47000);