#501736

order	512
nilpotency class	4
isoclinism	isoclinic class of #500736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24812
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

492 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408932);