#146885

order	512
nilpotency class	4
isoclinism	isoclinic class of #128584
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10806
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 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)(3,14,18,32)(4,19,22,52)(5,10)(6,26)(7,13)(8,16,31,47)(9,21,34,53)(11,24)(12,25)(15,41)(17,30,49,60)(20,51,23,37)(27,35,54,33)(28,36)(29,48)(38,55)(39,61)(40,62,56,46)(42,63)(43,57)(44,50)(45,59)(58,64);(1,3,5,16,7,18,25,47)(2,8,12,32,13,31,10,14)(4,15,20,42,22,45,23,44)(6,17,24,46,28,49,55,62)(9,29,35,57,34,58,33,39)(11,30,26,56,38,60,36,40)(19,50,37,59,52,63,51,41)(21,43,27,48,53,61,54,64);(1,4,5,20,7,22,25,23)(2,9,10,33,13,34,12,35)(3,15,16,42,18,45,47,44)(6,27,24,53,28,54,55,21)(8,29,14,39,31,58,32,57)(11,37,36,19,38,51,26,52)(17,48,46,61,49,64,62,43)(30,59,40,50,60,41,56,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54081);