#148734

order	512
nilpotency class	4
isoclinism	isoclinic class of #136628
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11208
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

465 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,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,h^a=hi,c^b=cf,g^b=gi,e^c=eh,f^c=fi,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, 55930);