#411391

order	512
nilpotency class	3
isoclinism	isoclinic class of #411390
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15115
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

708 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318587);