#406158

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14925
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 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)(3,15,19,37)(4,20,24,38)(5,11,26,36)(6,12)(7,13)(8,14)(9,18,32,27)(10,23,34,28)(16,30)(17,41)(21,49)(22,51)(25,35,29,39)(31,46)(33,52)(40,43)(42,61,50,55)(44,56)(45,58)(47,57)(48,54)(53,59)(60,62,64,63);(1,3,7,27,8,19,6,18)(2,9,12,15,14,32,13,37)(4,21,23,44,24,16,28,43)(5,17,29,46,26,47,25,45)(10,30,38,56,34,49,20,40)(11,31,35,41,36,58,39,57)(22,50,52,63,54,42,53,62)(33,55,51,64,59,61,48,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,28)(8,24)(9,30)(11,33)(12,20)(13,38)(14,34)(15,40)(17,42)(18,43)(19,21)(25,52)(26,54)(27,44)(29,53)(31,55)(32,49)(35,48)(36,59)(37,56)(39,51)(41,60)(45,62)(46,63)(47,50)(57,64)(58,61);(1,5,6,25,8,26,7,29)(2,11,12,35,14,36,13,39)(3,17,18,45,19,47,27,46)(4,22,23,52,24,54,28,53)(9,31,15,41,32,58,37,57)(10,33,20,48,34,59,38,51)(16,42,43,62,21,50,44,63)(30,55,40,60,49,61,56,64)

Polycyclic presentation:

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

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, 313354);