#486339

order	512
nilpotency class	4
isoclinism	isoclinic class of #486338
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18194
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

870 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)(3,13)(4,18)(5,23)(6,28)(7,12)(8,15)(9,22)(10,27)(11,26)(14,42)(16,48)(17,41)(19,40)(20,49)(21,54)(24,38)(25,39)(29,36)(30,52)(31,47)(32,57)(33,45)(34,43)(35,53)(37,46)(44,62)(50,56)(51,60)(55,59)(58,64)(61,63);(1,3,7,17)(2,8,12,31)(4,19,20,51)(5,24,25,57)(6,29,26,16)(9,33,34,59)(10,36,37,48)(11,38,28,32)(13,39,41,23)(14,43,44,22)(15,46,47,27)(18,30,49,58)(21,55,53,45)(35,60,54,40)(42,61,62,50)(52,63,64,56);(4,20)(5,25)(9,34)(14,44)(15,47)(18,49)(19,51)(22,43)(24,57)(27,46)(30,58)(33,59)(35,54)(40,60)(42,62)(50,61);(1,4,6,22,7,20,26,43)(2,9,11,18,12,34,28,49)(3,14,16,19,17,44,29,51)(5,21,27,56,25,53,46,63)(8,30,32,33,31,58,38,59)(10,35,23,50,37,54,39,61)(13,40,36,42,41,60,48,62)(15,45,24,52,47,55,57,64)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250769, #72250797, #72250804, …
maximal normal subgroup of	#72250769, #72250797, #72250804, …
maximal quotient of	#72250965

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393535);