#463481

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16473
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=f,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,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, 370677);