#148754

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,3)(5,6,11,17,12,19,34,18)(7,13,23,40,25,41,24,52)(8,26,30,48,32,51,31,44)(9,33,10,15,35,16,36,49)(14,42,45,64,46,29,53,28)(20,54,56,43,57,68,60,67)(21,47,58,50,59,39,22,38)(27,61,62,66,63,65,55,37);(1,2,4,3)(5,7)(6,13)(8,27)(9,22)(10,21)(11,23)(12,25)(14,43)(15,47)(16,50)(17,40)(18,52)(19,41)(20,29)(24,34)(26,61)(28,56)(30,62)(31,55)(32,63)(33,38)(35,58)(36,59)(37,44)(39,49)(42,57)(45,68)(46,67)(48,66)(51,65)(53,54)(60,64);(1,3,4,2)(5,8)(6,14)(7,20)(9,29)(10,28)(11,30)(12,32)(13,37)(15,48)(16,51)(17,45)(18,53)(19,46)(21,27)(22,55)(23,56)(24,60)(25,57)(26,33)(31,34)(35,42)(36,64)(38,67)(39,68)(40,61)(41,66)(43,50)(44,49)(47,54)(52,65)(58,62)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,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, 55950);