#149240

order	512
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2088
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11327
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,h^c=hi

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, 56436);