#497427

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

940 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)(3,9,16,36)(4,10)(5,22)(6,13)(7,28)(8,15,33,51)(11,26)(12,27)(14,35,47,63)(17,42,29,37)(18,54,55,62)(19,57)(20,41)(21,46)(23,45,52,34)(24,60)(25,31)(30,40)(32,50,61,56)(38,48,64,53)(39,59)(43,58)(44,49);(1,3)(2,8)(4,18)(5,23)(6,16)(7,29)(9,31)(10,38)(11,42)(12,45)(13,33)(14,19)(15,49)(17,27)(20,55)(21,50)(22,36)(24,48)(25,52)(26,51)(28,34)(30,56)(32,39)(35,40)(37,44)(41,64)(43,54)(46,63)(47,58)(53,59)(57,62)(60,61);(1,3)(2,9)(4,14)(5,15)(6,16)(7,17)(8,31)(10,35)(11,34)(12,37)(13,36)(18,19)(20,47)(21,48)(22,33)(23,49)(24,50)(25,51)(26,52)(27,29)(28,42)(30,53)(32,39)(38,40)(41,63)(43,62)(44,45)(46,64)(54,57)(55,58)(56,59)(60,61);(1,4,6,20)(2,10,13,41)(3,14,16,47)(5,24,25,59)(7,30,27,21)(8,32,33,61)(9,35,36,63)(11,43,44,57)(12,46,28,40)(15,50,51,56)(17,53,29,48)(18,52,55,23)(19,26,58,49)(22,60,31,39)(34,62,45,54)(37,64,42,38)

Polycyclic presentation:

2^9|a^2=g,d^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404623);