#131351

order	512
nilpotency class	4
isoclinism	isoclinic class of #131350
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

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

normal subgroups only

This group has 727 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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=eg,f^b=fgh,h^b=hi,d^c=dgi,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, 38547);