#149131

order	512
nilpotency class	4
isoclinism	isoclinic class of #140203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1934
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

409 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)(3,18,22,47,24,50,23,39)(4,25,30,43,32,57,31,44)(5,33,6,12,7,13,37,14)(10,34,40,36,41,21,49,20)(11,29,45,28,46,27,48,60)(19,51,53,62,54,38,55,56)(26,58,52,64,59,63,61,42);(1,3)(2,10,17,41)(4,26)(5,34)(6,36)(7,21)(8,22)(9,24)(11,42,46,64)(12,39,14,47)(13,18,33,50)(15,40,16,49)(19,29)(20,37)(23,35)(25,56,57,62)(27,54)(28,53)(30,52)(31,61)(32,59)(38,43,51,44)(45,58,48,63)(55,60);(1,4,5,27,35,30,37,60,9,32,7,29,8,31,6,28)(2,11,12,43,15,48,13,25,17,46,14,44,16,45,33,57)(3,19,20,26,22,55,34,61,24,54,36,59,23,53,21,52)(10,38,18,42,49,56,39,58,41,51,50,64,40,62,47,63)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,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=fh,h^b=hi,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, 56327);