#141408

order	512
nilpotency class	4
isoclinism	isoclinic class of #141343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=ehi,d^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48604);