#497869

order	512
nilpotency class	4
isoclinism	isoclinic class of #497868
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#22670
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3414	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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)(3,15,19,33)(4,21)(5,11)(6,10)(7,32)(8,14)(9,16,36,50)(12,24)(13,22)(17,47,53,61)(18,48,20,45)(23,55)(25,58,60,63)(26,43)(27,42)(28,44)(29,35,51,37)(30,38)(31,39)(34,49,62,59)(40,52,64,54)(41,56)(46,57);(1,3)(2,9)(4,16)(5,25)(6,29)(7,18)(8,19)(10,33)(11,40)(12,45)(13,35)(14,36)(15,39)(17,23)(20,22)(21,48)(24,50)(26,59)(27,49)(28,60)(30,52)(31,51)(32,37)(34,41)(38,62)(42,61)(43,47)(44,64)(46,58)(53,57)(54,56)(55,63);(1,4,22,31,8,24,7,6)(2,10,32,12,14,39,13,21)(3,16,20,51,19,50,18,29)(5,23,27,56,28,57,26,30)(9,33,37,45,36,15,35,48)(11,38,43,46,44,41,42,55)(17,49,54,60,53,59,52,25)(34,61,63,40,62,47,58,64);(1,5)(2,11)(3,17)(4,23)(6,30)(7,26)(8,28)(9,34)(10,38)(12,46)(13,42)(14,44)(15,47)(16,49)(18,52)(19,53)(20,54)(21,55)(22,27)(24,57)(25,29)(31,56)(32,43)(33,61)(35,58)(36,62)(37,63)(39,41)(40,45)(48,64)(50,59)(51,60)

Polycyclic presentation:

2^9|a^2=g,c^2=fi,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,e^d=ehi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285521, #72285570, #72285848
maximal normal subgroup of	#72285521, #72285570, #72285848
maximal quotient of	#72287426, #72287506, #72287597, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405065);