#148852

order	512
nilpotency class	4
isoclinism	isoclinic class of #141878
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1988
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11227
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

475 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,e^c=eh,g^c=gh,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, 56048);