#134516

order	512
nilpotency class	4
isoclinism	isoclinic class of #124849
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 373 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41712);