#416853

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15234
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 324049);