#144656

order	512
nilpotency class	4
isoclinism	isoclinic class of #144656
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10392
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,f^c=fi,e^d=ei

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51852);