#149383

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	#2076
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

444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=gh,h^c=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, 56579);