#486085

order	512
nilpotency class	4
isoclinism	isoclinic class of #486084
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18186
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#2190	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

630 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)(3,15)(4,21)(5,11)(6,30)(7,33)(8,14)(9,18)(10,24)(12,22)(13,27)(16,36)(17,49)(19,58)(20,50)(23,43)(25,41)(26,59)(28,62)(29,40)(31,46)(32,47)(34,44)(35,56)(37,57)(38,51)(39,55)(42,61)(45,60)(48,52)(53,64)(54,63);(1,3,8,20)(2,9,14,37)(4,16,24,52)(5,26,29,56)(6,31,32,51)(7,34,27,19)(10,35,21,59)(11,42,40,64)(12,44,39,58)(13,46,33,38)(15,47,50,30)(17,23,54,28)(18,55,57,22)(25,61,60,53)(36,62,48,43)(41,49,45,63);(1,4,8,24)(2,10)(3,16,20,52)(5,22,29,55)(6,28,32,23)(7,25,27,60)(9,35)(11,39)(12,40)(13,41)(14,21)(15,48)(17,51,54,31)(18,56,57,26)(19,53,34,61)(30,62)(33,45)(36,50)(37,59)(38,63)(42,58)(43,47)(44,64)(46,49);(1,5,27,23,8,29,7,28)(2,11,13,43,14,40,33,62)(3,17,34,56,20,54,19,26)(4,22,60,6,24,55,25,32)(9,36,38,42,37,48,46,64)(10,39,41,47,21,12,45,30)(15,49,44,35,50,63,58,59)(16,51,61,57,52,31,53,18)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250033, #72281886, #72307452
maximal normal subgroup of	#72250033, #72281886, #72307452
maximal quotient of	#72250333, #72279928

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393281);