#501701

order	512
nilpotency class	4
isoclinism	isoclinic class of #501701
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24780
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

596 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408897);