#136922

order	512
nilpotency class	4
isoclinism	isoclinic class of #136889
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

387 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=g,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44118);