#509786

order	512
nilpotency class	4
isoclinism	isoclinic class of #509783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31000
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33539	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

662 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416982);