#512640

order	512
nilpotency class	5
isoclinism	isoclinic class of #512640
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#3500	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×D₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

805 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)(3,13,16,37)(4,9)(5,21,23,52)(6,25,27,11)(7,12)(8,15,29,43)(10,24)(14,36,42,59)(17,39,26,38)(18,32)(19,47,45,63)(20,50,46,34)(22,30,44,31)(28,41,55,61)(33,49)(35,53)(40,58,51,60)(48,56,62,54)(57,64);(1,3)(2,8,12,31)(4,14)(5,22)(6,26)(7,17)(9,28,32,54)(10,13,35,39)(11,29,25,30)(15,23)(16,27)(18,40)(19,48)(20,51)(21,37,52,38)(24,44)(33,36,57,58)(34,55,50,56)(41,45)(42,46)(43,53)(47,59,63,60)(49,62)(61,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,28)(10,33)(11,34)(12,32)(13,36)(15,41)(16,42)(17,40)(21,47)(22,48)(23,45)(24,49)(25,50)(26,51)(27,46)(29,55)(30,56)(31,54)(35,57)(37,59)(38,60)(39,58)(43,61)(44,62)(52,63)(53,64);(4,18)(9,32)(14,40)(19,45)(20,46)(28,54)(33,57)(34,50)(36,58)(41,48)(42,51)(47,63)(49,64)(55,56)(59,60)(61,62)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419836);