#501660

order	512
nilpotency class	4
isoclinism	isoclinic class of #501658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24750
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

627 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=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, 408856);