#499780

order	512
nilpotency class	4
isoclinism	isoclinic class of #499777
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23848
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1245 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)(3,17)(4,24)(5,28)(6,31)(7,15)(8,14)(9,16)(10,20)(11,27)(12,29)(13,35)(18,51)(19,38)(21,57)(22,47)(23,50)(25,59)(26,46)(30,43)(32,49)(33,45)(34,42)(36,48)(37,54)(39,55)(40,56)(41,52)(44,62)(53,64)(58,60)(61,63);(1,3)(2,10)(4,25)(5,19)(6,32)(7,21)(8,22)(9,23)(11,41)(12,38)(13,47)(14,39)(15,49)(16,40)(17,31)(18,27)(20,35)(24,58)(26,61)(28,37)(29,52)(30,53)(33,54)(34,56)(36,55)(42,57)(43,63)(44,51)(45,64)(46,59)(48,50)(60,62);(1,4,9,5)(2,11,16,43)(3,18,23,53)(6,26,36,29)(7,30,8,27)(10,37,40,58)(12,42,46,13)(14,28,15,24)(17,38,50,59)(19,21,25,22)(20,52,56,61)(31,44,48,45)(32,60,55,54)(33,35,62,34)(39,63,49,41)(47,64,57,51);(1,5,9,4)(2,12,16,46)(3,19,23,25)(6,33,36,62)(7,30,8,27)(10,38,40,59)(11,42,43,13)(14,44,15,45)(17,37,50,58)(18,21,53,22)(20,52,56,61)(24,31,28,48)(26,35,29,34)(32,54,55,60)(39,51,49,64)(41,57,63,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406976);