#416863

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15235
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

631 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:

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

Polycyclic presentation:

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

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, 324059);