#136554

order	512
nilpotency class	4
isoclinism	isoclinic class of #136514
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#450
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=g,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dgi,e^b=egi,d^c=dgi,e^c=ehi,f^c=fi,h^c=hi

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, 43750);