#131437

order	512
nilpotency class	4
isoclinism	isoclinic class of #131436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8404
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 615 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fg,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, 38633);