#140981

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 82 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)(4,8)(5,6)(9,10)(11,13,22,19)(12,23,28,38)(14,17)(15,18)(16,27,31,37)(20,35)(21,29,30,34)(24,39)(25,32)(26,40)(33,36);(1,3,7,2,5,8,4,6)(9,11,19,18,15,22,13,10)(12,24,26,31,28,20,33,16)(14,30,34,17,25,21,29,32)(23,37,40,39,38,27,36,35);(2,6)(3,8)(9,12,25,37,15,28,14,27)(10,16,17,23,18,31,32,38)(11,20,21,36,22,24,30,40)(13,26,34,39,19,33,29,35)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48177);