#512654

order	512
nilpotency class	5
isoclinism	isoclinic class of #512651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#60831	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3499 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

791 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)(3,15,18,45)(4,10)(5,11)(6,27)(7,13,30,40)(8,14)(9,17,33,51)(12,29,42,26)(16,44,48,57)(19,32,20,46)(21,53,49,60)(22,35)(23,39)(24,37,56,61)(25,38)(28,43,52,34)(31,50,58,64)(36,55,41,54)(47,62,63,59);(1,3)(2,9)(4,21)(5,17)(6,28)(7,18)(8,20)(10,31)(11,32)(12,15)(13,43)(14,34)(16,25)(19,30)(22,50)(23,47)(24,49)(26,51)(27,45)(29,52)(33,40)(35,57)(36,53)(37,62)(38,59)(39,60)(41,44)(42,46)(48,56)(54,64)(55,63)(58,61);(1,4,8,25)(2,10,14,38)(3,16,20,21)(5,23,6,22)(7,24,30,56)(9,31,34,59)(11,39,27,35)(12,41,42,36)(13,37,40,61)(15,44,46,53)(17,50,28,47)(18,48,19,49)(26,55,29,54)(32,60,45,57)(33,58,43,62)(51,64,52,63);(1,5,8,6)(2,11,14,27)(3,17,20,28)(4,22,25,23)(7,26,30,29)(9,32,34,45)(10,35,38,39)(12,40,42,13)(15,33,46,43)(16,47,21,50)(18,51,19,52)(24,54,56,55)(31,57,59,60)(36,61,41,37)(44,62,53,58)(48,63,49,64)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi

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, 419850);