#131957

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	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

569 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,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, 39153);