#497208

order	512
nilpotency class	3
isoclinism	isoclinic class of #384259
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2333
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22339
prime radical	#3054	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,d^b=dh,f^b=fi,d^c=di,g^d=gi

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, 404404);