#130034

order	512
nilpotency class	4
isoclinism	isoclinic class of #129789
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#60714	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 636 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,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, 37230);