#147208

order	512
nilpotency class	4
isoclinism	isoclinic class of #13156
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1936
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10881
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

487 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,e^c=ei,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, 54404);