#146677

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
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 ◁ C₂×#568 ◁ 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)(3,15,19,31)(4,21,23,37)(5,12,6,11)(7,28)(8,14)(9,17)(10,22)(13,29)(16,41)(18,33)(20,49,48,40)(24,53,52,50)(25,36,27,38)(26,35)(30,44)(32,47)(34,54)(39,46,58,42)(43,57)(45,55)(51,56)(59,61,64,63)(60,62);(1,3,7,20,8,19,29,48)(2,9,13,32,14,33,28,55)(4,16,24,43,23,44,52,62)(5,17,25,45,6,18,27,47)(10,30,34,57,35,41,56,60)(11,15,36,40,12,31,38,49)(21,39,50,59,37,58,53,64)(22,42,51,61,26,46,54,63);(1,4)(2,10)(3,16)(5,22)(6,26)(7,24)(8,23)(9,30)(11,21)(12,37)(13,34)(14,35)(15,39)(17,42)(18,46)(19,44)(20,43)(25,51)(27,54)(28,56)(29,52)(31,58)(32,57)(33,41)(36,50)(38,53)(40,59)(45,61)(47,63)(48,62)(49,64)(55,60)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,f^b=fhi,e^c=ei,f^c=fhi,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, 53873);