#149387

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=h,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi,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, 56583);