#140979

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄×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	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 477 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)(12,23,27,38)(14,17)(15,18)(16,26,32,37)(20,35)(21,28,29,34)(24,39)(25,40)(30,31)(33,36);(1,3,7,2,5,8,4,6)(9,11,19,18,15,22,13,10)(12,24,25,32,27,20,33,16)(14,29,34,17,30,21,28,31)(23,37,40,39,38,26,36,35);(2,6)(3,8)(9,12,14,26,15,27,30,37)(10,16,31,38,18,32,17,23)(11,20,29,40,22,24,21,36)(13,25,28,35,19,33,34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,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, 48175);