#149148

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11316
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,f^c=fi

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, 56344);