#139805

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	#23924	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#537 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 717 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)(9,10,14,19,15,20,13,30)(11,21,25,37,26,38,31,33)(12,27,28,17,29,18,22,36)(16,32,34,23,35,39,40,24);(1,2)(3,5)(4,6)(7,8)(9,11,22,39)(10,16,17,33)(12,23,13,25)(14,31,29,24)(15,26,28,32)(18,21,30,40)(19,34,27,38)(20,35,36,37);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,23)(13,22)(14,28)(15,29)(16,33)(18,19)(20,36)(21,34)(24,26)(25,39)(27,30)(31,32)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,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, 47001);