#148393

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dhi,e^c=ei,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, 55589);