#148686

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11202
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

495 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55882);