#142644

order	512
nilpotency class	4
isoclinism	isoclinic class of #142644
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
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

622 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 68 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)(4,6)(7,8)(9,10)(11,22)(12,17)(13,20)(14,19)(15,21)(16,24)(18,29)(23,39)(25,36)(26,32)(27,33)(28,34)(30,37)(31,38)(35,40);(5,8)(9,11)(10,16)(12,28)(13,23)(14,26)(15,27)(17,22)(18,32)(19,35)(20,34)(21,36)(24,29)(25,30)(31,40)(33,38)(37,39);(1,3)(2,5)(4,7)(6,8)(10,17)(11,23)(12,29)(13,14)(16,32)(18,20)(19,37)(21,38)(22,35)(24,25)(26,27)(30,31)(33,34)(36,39)

Polycyclic presentation:

2^9|d^2=gi,e^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49840);