#143820

order	512
nilpotency class	4
isoclinism	isoclinic class of #143819
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28487	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

468 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=ei,d^c=dgi,e^c=eh,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51016);