#486700

order	512
nilpotency class	4
isoclinism	isoclinic class of #486696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18203
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#62330	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=gi,e^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250535, #72250556, #72250797, …
maximal normal subgroup of	#72250535, #72250556, #72250797, …
maximal quotient of	#72250929

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393896);