#480439

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
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 ◁ C₄×C₈ ◁ C₄×#131 ◁ 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)(3,14,17,39,18,31,44,54,15,8,26,29,41,27,50,52)(4,10,20,36,21,37,48,60,22,35,47,59,49,32,55,57)(16,34,42,58,43,56,61,64,19,28,45,53,46,51,62,63);(1,3)(2,8)(4,19)(5,17)(6,18)(7,15)(9,31)(10,34)(11,14)(12,39)(13,27)(16,22)(20,45)(21,46)(23,26)(24,44)(25,41)(28,35)(29,38)(30,50)(32,51)(33,52)(36,53)(37,56)(40,54)(42,47)(43,49)(48,62)(55,61)(57,64)(58,59)(60,63);(2,9,11,13)(3,15)(5,23)(8,27,14,31)(10,32,35,37)(12,40,38,33)(16,19)(18,41)(20,47)(24,30)(28,51,34,56)(29,54,39,52)(36,60,59,57)(43,46)(48,55)(53,64,58,63);(1,4,6,21,7,22,25,49)(2,10,13,37,11,35,9,32)(3,16,18,43,15,19,41,46)(5,20,24,48,23,47,30,55)(8,28,27,51,14,34,31,56)(12,36,40,60,38,59,33,57)(17,42,44,61,26,45,50,62)(29,53,52,63,39,58,54,64)

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,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, 387635);