#144790

order	512
nilpotency class	4
isoclinism	isoclinic class of #144790
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10397
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

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, 51986);