#146502

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, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#10726
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

809 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)(3,13,16,34)(4,18)(5,22,23,10)(6,25)(7,12)(8,15)(9,20)(11,26)(14,36,38,33)(17,43,42,35)(19,30,47,45)(21,31)(24,32,51,50)(27,53)(28,55)(29,41)(37,44)(39,57,61,59)(40,52)(46,49)(48,58,63,56)(54,60)(62,64);(1,3,6,17,7,16,26,42)(2,8,11,28,12,29,25,52)(4,14,21,39,20,38,49,61)(5,15,24,40,23,41,51,55)(9,27,31,54,18,44,46,62)(10,13,32,35,22,34,50,43)(19,37,48,60,47,53,63,64)(30,33,56,57,45,36,58,59);(1,4)(2,9,12,18)(3,14)(5,19)(6,21)(7,20)(8,27,29,44)(10,30,22,45)(11,31,25,46)(13,33,34,36)(15,37)(16,38)(17,39)(23,47)(24,48)(26,49)(28,54,52,62)(32,56,50,58)(35,57,43,59)(40,60)(41,53)(42,61)(51,63)(55,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,f^b=fhi,e^c=ei,f^d=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, 53698);