#138184

order	512
nilpotency class	4
isoclinism	isoclinic class of #138183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9136
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
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

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

normal subgroups only

This group has 301 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) = 64
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=ghi,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=fhi,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, 45380);