#142832

order	512
nilpotency class	4
isoclinism	isoclinic class of #142831
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33520	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50028);