#502202

order	512
nilpotency class	4
isoclinism	isoclinic class of #502201
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25148
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

723 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409398);