#138080

order	512
nilpotency class	4
isoclinism	isoclinic class of #138078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9118
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45276);