#131926

order	512
nilpotency class	4
isoclinism	isoclinic class of #131824
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#61284	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

596 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=egi,f^c=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, 39122);