#147386

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#388
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

677 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^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,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, 54582);