#497428

order	512
nilpotency class	4
isoclinism	isoclinic class of #497426
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#22436
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3416	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

948 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)(3,13,17,40)(4,19,21,9)(5,23)(6,27)(7,11)(8,16,32,48)(10,25)(12,29)(14,42)(15,38,47,33)(18,31,28,41)(20,52,54,58)(22,35,56,51)(24,37)(26,36)(30,44)(34,55,50,53)(39,46)(43,60)(45,62)(49,64)(57,59)(61,63);(1,3)(2,8)(4,14)(5,15)(6,28)(7,17)(9,30)(10,31)(11,32)(12,38)(13,36)(16,24)(18,29)(19,49)(20,43)(21,46)(22,57)(23,40)(25,48)(26,47)(27,33)(34,59)(35,61)(37,41)(39,52)(42,58)(44,53)(45,56)(50,62)(51,60)(54,63)(55,64);(1,4,7,21)(2,9,11,19)(3,14,17,46)(5,20,26,54)(6,22,29,56)(8,30,32,49)(10,34,37,50)(12,35,27,51)(13,39,40,42)(15,43,47,63)(16,44,48,64)(18,45,28,57)(23,58,36,52)(24,53,25,55)(31,59,41,62)(33,60,38,61);(3,15)(5,24)(6,29)(8,31)(10,36)(12,27)(13,38)(14,43)(16,28)(17,47)(18,48)(20,53)(22,56)(23,37)(25,26)(30,59)(32,41)(33,40)(34,52)(35,51)(39,61)(42,60)(44,57)(45,64)(46,63)(49,62)(50,58)(54,55)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=h,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,g^c=gi,e^d=ehi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319782
maximal normal subgroup of	#72319782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404624);