#149154

order	512
nilpotency class	4
isoclinism	isoclinic class of #140148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1926
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11316
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

413 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)(3,15)(4,20,25,40)(5,27)(6,30)(7,33)(8,14)(9,17)(10,24,41,23)(11,22)(12,32)(13,29)(16,47,49,62)(18,44)(19,36)(21,39,56,46)(26,60,58,54)(28,45)(31,37)(34,43)(35,55,52,50)(38,51)(42,48,53,59)(57,64,61,63);(1,3,7,34,8,19,29,18)(2,9,33,37,14,38,13,45)(4,21,26,61,25,56,58,57)(5,28,22,17,6,31,32,51)(10,39,53,63,41,46,42,64)(11,43,30,36,12,44,27,15)(16,48,55,24,49,59,50,23)(20,52,60,62,40,35,54,47);(1,4,22,48,7,26,6,24,8,25,32,59,29,58,5,23)(2,10,27,54,13,42,12,40,14,41,30,60,33,53,11,20)(3,16,28,57,18,50,51,56,19,49,31,61,34,55,17,21)(9,35,43,64,37,47,36,39,38,52,44,63,45,62,15,46)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,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, 56350);