#147367

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10921
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

705 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54563);