#141266

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9735
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

280 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48462);