#479763

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17237
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

540 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386959);