#499043

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2034
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1047 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)(4,21)(5,11)(6,27)(7,13)(8,14)(9,18)(10,24)(12,30)(16,46)(17,36)(19,50)(20,34)(22,57)(23,55)(25,56)(26,44)(28,60)(29,38)(31,43)(32,42)(33,62)(35,49)(37,51)(39,61)(40,59)(41,52)(45,48)(47,64)(53,54)(58,63);(1,3,7,20,8,18,5,17)(2,9,11,34,14,15,13,36)(4,22,23,47,25,53,32,16)(6,28,30,49,31,19,26,52)(10,37,39,45,40,63,38,33)(12,41,27,50,44,35,43,60)(21,54,42,64,56,57,55,46)(24,58,29,48,59,51,61,62);(1,4)(2,10)(3,16,18,47)(5,23)(6,29)(7,32)(8,25)(9,33,15,45)(11,38)(12,42)(13,39)(14,40)(17,22,20,53)(19,51,28,58)(21,27)(24,30)(26,59)(31,61)(34,63,36,37)(35,57,41,54)(43,56)(44,55)(46,60,64,50)(48,49,62,52);(1,5,8,7)(2,11,14,13)(3,17,18,20)(4,23,25,32)(6,26,31,30)(9,34,15,36)(10,38,40,39)(12,27,44,43)(16,22,47,53)(19,49,28,52)(21,55,56,42)(24,29,59,61)(33,63,45,37)(35,60,41,50)(46,57,64,54)(48,51,62,58)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320498, #72320647, #72320724
maximal normal subgroup of	#72320498, #72320647, #72320724
maximal quotient of	#72322772

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406239);