#509180

order	512
nilpotency class	4
isoclinism	isoclinic class of #509179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30444
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

766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416376);