#131357

order	512
nilpotency class	4
isoclinism	isoclinic class of #131354
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8403
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60773	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

723 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 723 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=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, 38553);