#142647

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
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 580 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,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, 49843);