#141998

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 102 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)(9,10,14,19,15,20,31,18)(11,21,23,13,24,29,17,30)(12,25,27,16,28,33,40,34)(22,36,37,35,38,26,32,39);(2,5)(6,7)(9,11,13,20,31,17,29,18,15,24,30,10,14,23,21,19)(12,22,26,34,40,32,39,25,28,38,36,16,27,37,35,33);(1,3,4,8)(2,6,5,7)(9,12,14,27,15,28,31,40)(10,16,18,25,20,34,19,33)(11,22,23,37,24,38,17,32)(13,26,21,35,30,36,29,39)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,d^2=ghi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,f^b=fh,g^b=gi,h^b=hi,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, 49194);