#131928

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	#61287	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

586 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=hi,e^2=gi,f^2=ghi,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, 39124);