#147615

order	512
nilpotency class	4
isoclinism	isoclinic class of #147614
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10966
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

861 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)(4,20,24,36)(5,25,28,11)(6,12)(7,29)(8,13)(9,16,33,48)(10,23,37,30)(15,43,45,62)(17,41)(18,49)(19,42)(21,53,50,40)(22,55)(26,59,27,38)(31,54)(32,46,39,47)(34,44)(35,56)(51,64)(52,57)(58,60)(61,63);(1,3,8,19)(2,9,13,33)(4,21,24,50)(5,16,28,48)(6,18,7,17)(10,34,37,61)(11,14,25,42)(12,39,29,32)(15,30,45,23)(20,51,36,31)(22,54,57,64)(26,47,27,46)(35,62,58,43)(38,49,59,41)(40,52,53,55)(44,60,63,56);(1,4)(2,10,13,37)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31,33,51)(11,35,25,58)(12,20,29,36)(14,40,42,53)(16,44)(17,21)(18,50)(19,45)(26,56)(27,60)(28,57)(32,61,39,34)(38,52,59,55)(41,43,49,62)(46,54)(47,64)(48,63)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gh,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54811);