#145961

order	512
nilpotency class	4
isoclinism	isoclinic class of #145961
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10597
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
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

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

normal subgroups only

This group has 205 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:

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

Polycyclic presentation:

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

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, 53157);