#436754

order	512
nilpotency class	3
isoclinism	isoclinic class of #420517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15569
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

907 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)(3,14)(4,19,24,46)(5,25)(6,30)(7,12)(8,13)(9,17)(10,23,38,22)(11,29)(15,32,20,36)(16,39)(18,34)(21,49,53,45)(26,33)(27,40)(28,56)(31,35)(37,51,55,52)(41,43,60,48)(42,59,47,62)(44,54)(50,64,57,63)(58,61);(1,3,7,17,8,18,6,31)(2,9,12,34,13,35,30,14)(4,20,23,48,24,15,22,43)(5,26,28,44,29,58,27,16)(10,36,19,41,38,32,46,60)(11,39,40,33,25,54,56,61)(21,50,52,42,53,57,51,47)(37,63,45,62,55,64,49,59);(1,4,6,22,8,24,7,23)(2,10,12,19,13,38,30,46)(3,15,17,43,18,20,31,48)(5,21,27,51,29,53,28,52)(9,32,14,41,35,36,34,60)(11,37,40,45,25,55,56,49)(16,42,26,57,44,47,58,50)(33,59,39,64,61,62,54,63);(1,5)(2,11,13,25)(3,16,18,44)(4,21)(6,27)(7,28)(8,29)(9,33)(10,37,38,55)(12,40,30,56)(14,39)(15,42,20,47)(17,26,31,58)(19,45,46,49)(22,51)(23,52)(24,53)(32,59)(34,54)(35,61)(36,62)(41,64)(43,57,48,50)(60,63)

Polycyclic presentation:

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

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, 343950);