#145876

order	512
nilpotency class	4
isoclinism	isoclinic class of #143329
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1634
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10576
prime radical	C₂³	#456
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

221 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53072);