#134481

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1799
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,f^b=fi,e^c=eh,f^c=fghi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41677);