#480443

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	#17363
prime radical	#3455	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #97 ◁ #2875 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,d^b=di,f^b=fg,d^c=di,f^c=fi,f^e=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, 387639);