#486702

order	512
nilpotency class	4
isoclinism	isoclinic class of #486701
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	#62335	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

664 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,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,d^c=di,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250033, #72250052, #72250076, …
maximal normal subgroup of	#72250033, #72250052, #72250076, …
maximal quotient of	#72250350, #72279861

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393898);