#6573731

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

4403 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,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	#72342111
maximal subgroup of	#72342111, #72342140, #72343955
maximal normal subgroup of	#72342111, #72342140, #72343955
maximal quotient of	#72343248

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480927);