#413786

order	512
nilpotency class	3
isoclinism	isoclinic class of #403107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2547
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15156
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

675 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)(3,15)(4,21,26,10)(5,27)(6,14)(7,12)(8,13)(9,18)(11,29)(16,35,43,49)(17,44)(19,33)(20,34)(22,31,48,58)(23,52,55,56)(24,37,25,38)(28,40)(30,39)(32,46)(36,54,50,53)(41,45)(42,61,62,57)(47,59)(51,60,63,64);(1,3,6,20,8,19,7,18)(2,9,14,15,13,34,12,33)(4,22,24,16,26,48,25,43)(5,17,28,47,30,46,29,45)(10,35,38,58,21,49,37,31)(11,32,27,41,40,44,39,59)(23,51,53,42,55,63,54,62)(36,60,56,61,50,64,52,57);(1,4,6,24,8,26,7,25)(2,10,14,38,13,21,12,37)(3,16,20,48,19,43,18,22)(5,23,28,53,30,55,29,54)(9,31,15,35,34,58,33,49)(11,36,27,56,40,50,39,52)(17,42,47,63,46,62,45,51)(32,57,41,60,44,61,59,64);(1,5,7,29,8,30,6,28)(2,11,14,27,13,40,12,39)(3,17,18,45,19,46,20,47)(4,23,25,54,26,55,24,53)(9,32,15,41,34,44,33,59)(10,36,38,56,21,50,37,52)(16,42,22,51,43,62,48,63)(31,57,35,60,58,61,49,64)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,d^2=h,e^2=i,g^2=i,h^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,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, 320982);