#148386

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

664 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)(7,18,21,49)(8,14)(9,29)(10,33)(11,36)(12,17)(13,20,39,22)(15,32)(16,31)(19,51,53,48)(23,60,56,50)(24,62,52,67)(25,63)(26,45)(27,47)(28,44)(30,64)(34,42)(35,46)(37,43)(38,58,66,55)(40,59,65,57)(41,61,68,54);(5,7,9,20,11,23,32,57,12,21,10,22,31,56,35,59)(6,13,15,18,16,40,29,50,17,39,46,49,36,65,33,60)(8,24,25,61,27,55,34,19,28,52,26,54,37,58,30,53)(14,41,42,62,43,51,63,66,44,68,64,67,47,48,45,38);(2,4)(5,8,12,28)(6,14,17,44)(7,19,21,53)(9,30,10,34)(11,37,31,27)(13,38,39,66)(15,45,46,63)(16,47,36,43)(18,48,49,51)(20,55,22,58)(23,61,56,54)(24,59,52,57)(25,32,26,35)(29,64,33,42)(40,67,65,62)(41,50,68,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,f^2=h,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,f^c=fhi,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, 55582);