#146880

order	512
nilpotency class	4
isoclinism	isoclinic class of #128583
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10806
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54076);