#510220

order	512
nilpotency class	4
isoclinism	isoclinic class of #510219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31382
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28250	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#444 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 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)(4,20)(5,9)(6,29)(7,13)(8,17)(10,22)(11,27)(12,31)(15,44)(16,47)(18,34)(19,35)(21,53)(23,57)(24,40)(25,41)(26,36)(28,42)(30,43)(32,51)(33,48)(37,45)(38,50)(39,58)(46,61)(49,63)(52,59)(54,62)(55,64)(56,60);(1,3,5,26,7,17,27,18)(2,8,11,36,13,14,9,34)(4,21,22,48,25,16,24,50)(6,30,28,19,31,15,45,51)(10,38,20,47,40,33,41,53)(12,43,42,32,29,44,37,35)(23,46,56,49,59,54,58,55)(39,63,52,61,60,64,57,62);(2,9)(3,15,17,30)(5,27)(6,28)(8,32,14,35)(10,20)(11,13)(12,29)(16,46,21,54)(18,51,26,19)(22,24)(23,58)(31,45)(33,61,38,62)(34,44,36,43)(39,60)(40,41)(47,64,53,63)(48,49,50,55)(56,59);(1,4,7,25)(2,10,13,40)(3,16,17,21)(5,24,27,22)(6,23,31,59)(8,33,14,38)(9,20,11,41)(12,39,29,60)(15,46,30,54)(18,50,26,48)(19,49,51,55)(28,58,45,56)(32,61,35,62)(34,53,36,47)(37,52,42,57)(43,64,44,63)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417416);