#149169

order	512
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
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:

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

Polycyclic presentation:

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

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, 56365);