#142643

order	512
nilpotency class	4
isoclinism	isoclinic class of #142640
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

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

normal subgroups only

This group has 585 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)(9,10,14,19)(11,22,12,27)(13,28,30,39)(15,21,16,20)(17,24,33,23)(18,32,38,31)(25,37,26,36)(29,35,40,34);(5,8)(9,11)(10,17)(12,14)(13,29)(15,26)(16,25)(18,36)(19,33)(20,35)(21,34)(22,28)(23,31)(24,32)(27,39)(30,40)(37,38);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,17,19,33)(13,29,30,40)(15,25,16,26)(18,37,38,36)(20,35,21,34)(22,28,27,39)(23,32,24,31)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,e^2=gi,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,g^b=gi,h^b=hi,d^c=dh,e^c=eg,g^c=gi,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, 49839);