#486346

order	512
nilpotency class	4
isoclinism	isoclinic class of #486345
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

862 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)(3,17)(4,23)(5,28)(6,33)(7,15)(8,14)(9,16)(10,20)(11,26)(12,31)(13,24)(18,53)(19,38)(21,61)(22,52)(25,48)(27,46)(29,51)(30,43)(32,47)(34,40)(35,50)(36,44)(37,49)(39,55)(41,54)(42,60)(45,57)(56,64)(58,63)(59,62);(1,3,9,22)(2,10,16,42)(4,18,27,57)(5,29,32,59)(6,34,35,54)(7,21,8,37)(11,38,44,63)(12,45,48,53)(13,49,43,61)(14,41,15,40)(17,50,52,33)(19,25,58,31)(20,30,60,24)(23,51,46,62)(26,55,36,56)(28,39,47,64);(1,4)(2,11)(3,18)(5,30)(6,31)(7,36)(8,26)(9,27)(10,38)(12,13)(14,46)(15,23)(16,44)(17,39)(19,34)(20,59)(21,56)(22,57)(24,32)(25,35)(28,33)(29,60)(37,55)(40,51)(41,62)(42,63)(43,48)(45,49)(47,50)(52,64)(53,61)(54,58);(1,5,8,31,9,32,7,25)(2,12,15,28,16,48,14,47)(3,19,21,29,22,58,37,59)(4,24,26,35,27,30,36,6)(10,39,41,45,42,64,40,53)(11,43,23,50,44,13,46,33)(17,51,49,38,52,62,61,63)(18,54,56,20,57,34,55,60)

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gi,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=eg,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	#72250025, #72250033, #72250056, …
maximal normal subgroup of	#72250025, #72250033, #72250056, …
maximal quotient of	#72250258, #72279876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393542);