#146170

order	512
nilpotency class	4
isoclinism	isoclinic class of #146170
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10654
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

191 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 191 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=d,c^2=dh,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fh,g^c=gi,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, 53366);