#148520

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

638 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)(8,26,31,49)(9,33)(10,15)(11,37)(12,19)(13,22)(14,29,46,28)(16,36)(17,35)(18,34)(21,52,56,66)(23,50)(24,41)(25,40)(27,44,62,68)(30,60,53,47)(32,45,63,61)(38,48)(39,57,65,64)(42,59)(43,58)(51,55,67,54);(5,7,12,25)(6,13,19,43)(8,27,31,62)(9,23,35,59)(10,22,36,58)(11,38,34,24)(14,44,46,68)(15,48,16,41)(17,40,33,20)(18,50,37,42)(21,53,56,30)(26,52,49,66)(28,57,29,64)(32,54,63,55)(39,61,65,45)(47,67,60,51);(2,4)(5,8,9,28,34,63,36,53,12,31,35,29,11,32,10,30)(6,14,15,26,37,60,17,45,19,46,16,49,18,47,33,61)(7,21,22,54,38,64,59,62,25,56,58,55,24,57,23,27)(13,39,20,51,50,52,41,44,43,65,40,67,42,66,48,68)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=hi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,e^c=ei,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, 55716);