#138214

order	512
nilpotency class	4
isoclinism	isoclinic class of #138213
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9142
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33190	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

395 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=gh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,e^c=ei,f^c=fi

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, 45410);