#146742

order	512
nilpotency class	4
isoclinism	isoclinic class of #128584
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

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	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

935 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,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, 53938);