#149008

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11268
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

431 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)(3,16,21,42)(4,23,28,38)(5,29)(6,33)(7,13)(8,15)(9,19,36,30)(10,26,40,25)(11,32)(12,17)(14,31)(18,37)(20,45,22,52)(24,56)(27,54,59,55)(34,49)(35,51,43,46)(39,47,41,60)(44,61)(48,64)(50,62)(53,58)(57,63);(1,3,17,46,31,20,6,19,8,21,32,51,7,22,5,30)(2,9,33,45,13,43,12,16,15,36,29,52,14,35,11,42)(4,24,47,48,59,50,26,57,28,49,60,61,27,18,25,58)(10,37,55,64,39,34,23,53,40,62,54,44,41,56,38,63);(1,4,5,25,7,27,32,60,8,28,6,26,31,59,17,47)(2,10,11,38,14,41,29,54,15,40,12,23,13,39,33,55)(3,18,19,48,22,49,46,58,21,50,30,61,20,24,51,57)(9,34,16,44,35,37,45,53,36,56,42,64,43,62,52,63)

Polycyclic presentation:

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

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, 56204);