#148951

order	512
nilpotency class	4
isoclinism	isoclinic class of #140140
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1925
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11256
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

427 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^b=fh,h^b=hi,f^c=fhi,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, 56147);