#501255

order	512
nilpotency class	4
isoclinism	isoclinic class of #501252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24424
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61106	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,g^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,g^c=gi,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, 408451);