#149095

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2034
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

458 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)(5,6,11,16,12,17,35,48,13,18,9,33,10,37,36,50)(7,19,23,40,24,47,38,42,25,41,21,49,22,57,34,14)(8,26,30,58,31,46,64,65,32,59,28,63,29,15,43,45)(20,52,27,60,55,44,62,67,56,68,53,39,54,51,61,66);(5,7)(6,14,18,42)(8,27)(9,34)(10,24)(11,38)(12,22)(13,25)(15,44,46,51)(16,47,33,57)(17,49,37,40)(19,50,41,48)(20,28)(21,36)(23,35)(26,52,59,68)(29,62)(30,56)(31,61)(32,53)(39,65,60,45)(43,54)(55,64)(58,67,63,66);(5,8,9,28,12,31,36,43,13,32,11,30,10,29,35,64)(6,15,16,45,17,26,48,58,18,46,33,65,37,59,50,63)(7,20,21,53,24,55,34,61,25,56,23,27,22,54,38,62)(14,39,19,51,40,66,47,52,42,60,41,44,49,67,57,68)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56291);