#6573734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573729
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,h^b=hi,d^c=dh,f^c=fi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72342123
maximal subgroup of	#72342123, #72342140, #72343961
maximal normal subgroup of	#72342123, #72342140, #72343961
maximal quotient of	#72343245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480930);