#149140

order	512
nilpotency class	4
isoclinism	isoclinic class of #140140
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1925
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

413 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)(3,14,18,35)(4,20)(5,26)(6,29)(7,32)(8,13)(9,17,36,53)(10,23)(11,28)(12,15)(16,48,51,62)(19,42,33,46)(21,38,50,45)(22,41)(24,60)(25,39)(27,44,30,37)(31,43)(34,52,54,57)(40,49)(47,56)(55,59)(58,64,61,63);(1,3,15,19,8,18,7,33)(2,9,12,44,13,36,32,37)(4,21,47,58,25,50,24,61)(5,27,31,17,6,30,28,53)(10,38,40,64,41,45,55,63)(11,42,26,35,43,46,29,14)(16,49,52,22,51,59,57,23)(20,54,56,62,39,34,60,48);(1,4,5,22,7,24,28,49,8,25,6,23,15,47,31,59)(2,10,11,20,12,40,26,56,13,41,43,39,32,55,29,60)(3,16,17,50,19,52,30,61,18,51,53,21,33,57,27,58)(9,34,14,45,37,62,46,64,36,54,35,38,44,48,42,63)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,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, 56336);