#501864

order	512
nilpotency class	4
isoclinism	isoclinic class of #501864
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24909
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2192	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 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)(3,16)(4,22)(5,26)(6,15)(7,14)(8,13)(9,19)(10,24)(11,23)(12,31)(17,50)(18,35)(20,43)(21,49)(25,45)(27,36)(28,47)(29,38)(30,44)(32,40)(33,48)(34,54)(37,56)(39,58)(41,53)(42,61)(46,55)(51,57)(52,64)(59,63)(60,62);(1,3,6,20,7,21,8,33)(2,9,13,36,14,37,15,46)(4,17,24,18,25,53,32,52)(5,27,29,19,30,55,31,56)(10,34,22,57,40,63,45,62)(11,41,42,35,39,50,47,64)(12,43,44,16,38,48,26,49)(23,54,28,60,58,59,61,51);(1,4)(2,10)(3,17)(5,23)(6,32)(7,25)(8,24)(9,34)(11,38)(12,39)(13,45)(14,40)(15,22)(16,35)(18,33)(19,51)(20,52)(21,53)(26,47)(27,54)(28,31)(29,61)(30,58)(36,62)(37,63)(41,48)(42,44)(43,50)(46,57)(49,64)(55,59)(56,60);(1,4)(2,11)(3,18)(5,28)(6,24)(7,25)(8,32)(9,35)(10,38)(12,40)(13,42)(14,39)(15,47)(16,34)(17,33)(19,51)(20,53)(21,52)(22,26)(23,31)(27,59)(29,58)(30,61)(36,50)(37,64)(41,46)(43,62)(44,45)(48,57)(49,63)(54,55)(56,60)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409060);