#149382

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	#2092
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:

(1,2)(3,4)(5,6,12,17,13,19,10,18,14,20,9,35,38,52,11,39)(7,21,25,45,26,43,24,46,27,55,23,53,40,44,36,15)(8,28,32,51,33,64,31,54,34,49,30,16,41,48,37,50)(22,56,59,68,60,42,58,66,61,67,57,47,62,63,29,65);(2,4)(5,7)(6,15,20,46)(8,29)(9,36)(10,25)(11,23)(12,24)(13,40)(14,27)(16,47,51,68)(17,43,35,44)(18,21,39,55)(19,53,52,45)(22,31)(26,38)(28,63,49,42)(30,62)(32,60)(33,57)(34,58)(37,61)(41,59)(48,56,64,67)(50,66,54,65);(2,4)(5,8)(6,16,20,51)(7,22)(9,37)(10,32)(11,30)(12,31)(13,41)(14,34)(15,42,46,63)(17,28,35,49)(18,48,39,64)(19,54,52,50)(21,47,55,68)(23,29)(24,59)(25,58)(26,62)(27,61)(33,38)(36,57)(40,60)(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=h,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, 56578);