#501658

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	#33618	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,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, 408854);