#131435

order	512
nilpotency class	4
isoclinism	isoclinic class of #131432
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

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

normal subgroups only

This group has 611 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,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=egi,f^b=fgh,h^b=hi,d^c=dg,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, 38631);