#142633

order	512
nilpotency class	4
isoclinism	isoclinic class of #142632
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	C₂×#2184	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 70 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)(11,24)(12,17)(13,32)(14,22)(15,21)(16,23)(18,26)(19,29)(20,31)(25,35)(27,39)(28,34)(30,36)(33,37)(38,40);(5,8)(10,17)(11,25)(12,29)(13,31)(14,15)(18,35)(19,21)(20,22)(23,32)(24,37)(26,40)(27,28)(34,36)(38,39);(1,3)(2,5)(4,7)(6,8)(9,11,16,28)(10,18,23,38)(12,30,13,33)(14,27,15,25)(17,34,32,24)(19,35,20,39)(21,37,22,36)(26,29,40,31)

Polycyclic presentation:

2^9|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=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49829);