#147387

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10922
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

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

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, 54583);