#141377

order	512
nilpotency class	4
isoclinism	isoclinic class of #141376
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9764
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	#33740	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,13,36,11,8,15,24,46,37,14,47,30)(3,5,18,16,19,33,35,51,21,34,45,32,20,50,10,43)(4,22,27,9,28,39,58,40,29,59,26,42,62,48,63,41)(17,52,54,64,23,60,44,25,56,49,57,61,55,53,38,31);(1,3,16,13,37,19,43,15,8,21,32,14,7,20,51,2)(4,23,49,42,62,56,60,41,29,55,52,9,28,17,53,40)(5,30,6,35,50,12,47,45,34,11,24,10,33,46,36,18)(22,58,44,25,39,27,57,64,59,63,38,31,48,26,54,61);(1,4,24,27,7,28,47,58,8,29,6,26,37,62,36,63)(2,9,30,59,14,41,46,39,15,42,11,22,13,40,12,48)(3,17,35,38,20,55,18,57,21,56,10,44,19,23,45,54)(5,31,16,49,33,64,51,53,34,25,32,52,50,61,43,60)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,d^2=ghi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48573);