#148503

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
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

642 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)(7,20)(8,26,31,59)(9,33)(10,15)(11,37)(12,19)(13,22)(14,29,46,65)(16,35)(17,36)(18,34)(21,51,54,39)(23,50)(24,40)(25,41)(27,60,52,66)(28,63,30,47)(32,49,64,45)(38,48)(42,57)(43,56)(44,53,67,62)(55,58,61,68);(5,7,12,25)(6,13,19,43)(8,27,31,52)(9,23,35,57)(10,22,36,56)(11,38,34,24)(14,44,46,67)(15,48,17,40)(16,41,33,20)(18,50,37,42)(21,28,54,30)(26,58,59,68)(29,55,65,61)(32,53,64,62)(39,49,51,45)(47,66,63,60);(2,4)(5,8,9,28,34,64,36,65,12,31,35,30,11,32,10,29)(6,14,15,45,37,63,16,26,19,46,17,49,18,47,33,59)(7,21,22,52,38,61,57,62,25,54,56,27,24,55,23,53)(13,39,20,44,50,68,40,66,43,51,41,67,42,58,48,60)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=hi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,f^c=fi,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, 55699);