#149248

order	512
nilpotency class	4
isoclinism	isoclinic class of #13460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2058
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11327
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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)(3,17,22,49)(4,24)(5,14)(6,15)(7,32)(8,13)(9,16)(10,19,38,55)(11,27)(12,30)(18,50,53,64)(20,46,23,37)(21,39,33,36)(25,58,54,62)(26,47)(28,44)(29,48)(31,45)(34,43)(35,51,61,56)(40,57,63,52)(41,60)(42,59);(1,3,5,19,8,23,30,21,9,22,31,55,6,20,7,33)(2,10,12,17,15,39,14,46,16,38,32,49,13,36,45,37)(4,25,26,53,27,56,59,57,29,54,60,18,34,51,28,52)(11,40,41,35,24,50,42,58,43,63,47,61,48,64,44,62);(1,4,6,27,9,29,8,34)(2,11,13,24,16,43,15,48)(3,18,20,52,22,53,23,57)(5,28,7,26,31,59,30,60)(10,35,36,58,38,61,39,62)(12,44,45,41,32,42,14,47)(17,40,37,50,49,63,46,64)(19,54,33,51,55,25,21,56)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,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, 56444);