#138390

order	512
nilpotency class	4
isoclinism	isoclinic class of #138390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9157
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

345 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=gh,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,d^c=di,e^c=ei,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, 45586);