#6573738

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573735
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

4409 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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

71 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72342060, #72342104
maximal subgroup of	#72341995, #72342045, #72342060
maximal normal subgroup of	#72342045, #72342060

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480934);