#148908

order	512
nilpotency class	4
isoclinism	isoclinic class of #140408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1949
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
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

467 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)(4,22,27,57)(5,28)(6,12)(7,14)(8,33)(9,15)(10,18)(11,25,42,60)(13,30)(17,48,52,34)(19,36)(20,35)(21,47)(23,58,53,39)(24,41,49,43)(26,56,32,40)(29,44)(31,46)(37,55)(38,54)(45,51,63,59)(50,61,64,62);(1,3,6,19,8,20,30,54,9,21,29,55,7,31,5,18)(2,10,28,46,14,37,44,47,15,38,13,35,33,36,12,16)(4,23,25,50,26,17,49,51,27,53,60,64,32,52,24,59)(11,39,57,63,43,48,56,61,42,58,22,45,41,34,40,62);(1,4,5,24,7,32,29,60,9,27,30,49,8,26,6,25)(2,11,12,40,33,41,13,22,15,42,44,56,14,43,28,57)(3,17,18,50,31,23,55,59,21,52,54,64,20,53,19,51)(10,34,16,45,36,58,35,61,38,48,47,63,37,39,46,62)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,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,f^b=fi,f^c=fi,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, 56104);