#148776

order	512
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2085
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11219
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,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, 55972);