#510705

order	512
nilpotency class	4
isoclinism	isoclinic class of #510704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31851
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13310	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

482 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417901);