#501467

order	512
nilpotency class	4
isoclinism	isoclinic class of #501466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24607
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=e,c^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408663);