#149361

order	512
nilpotency class	4
isoclinism	isoclinic class of #135291
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1825
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11372
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

423 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,g^c=gh,h^c=hi

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, 56557);