#148772

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11219
prime radical	C₂⁴	#137
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,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, 55968);