#146682

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

993 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)(3,15,19,45)(4,21)(5,26,29,11)(6,30,32,12)(7,33)(8,14)(9,17)(10,23,39,57)(13,28)(16,46,49,63)(18,36)(20,42,53,35)(22,40,27,55)(24,38)(25,56)(31,44)(34,50,62,64)(37,52)(41,59)(43,58)(47,54)(48,60)(51,61);(1,3,7,20,8,19,28,53)(2,9,13,36,14,37,33,44)(4,16,24,50,25,49,59,64)(5,17,6,31,29,52,32,18)(10,34,40,63,39,62,55,46)(11,15,30,35,26,45,12,42)(21,54,41,43,56,61,38,60)(22,47,57,48,27,51,23,58);(1,4)(2,10)(3,16)(5,27)(6,23)(7,24)(8,25)(9,34)(11,41)(12,21)(13,40)(14,39)(15,43)(17,51)(18,48)(19,49)(20,50)(22,29)(26,38)(28,59)(30,56)(31,58)(32,57)(33,55)(35,61)(36,63)(37,62)(42,54)(44,46)(45,60)(47,52)(53,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,f^c=fh,f^d=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, 53878);