#148227

order	512
nilpotency class	4
isoclinism	isoclinic class of #134415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
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

670 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,g^c=gh,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, 55423);