#146176

order	512
nilpotency class	4
isoclinism	isoclinic class of #146176
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10655
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

183 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=d,c^2=d,d^2=i,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,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, 53372);