#141279

order	512
nilpotency class	4
isoclinism	isoclinic class of #141279
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9739
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,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,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,h^b=hi,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, 48475);