#141995

order	512
nilpotency class	4
isoclinism	isoclinic class of #129430
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1466
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=defg,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,f^b=fgi,g^b=gi,f^c=fi,f^d=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, 49191);