#149051

order	512
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2088
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

445 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56247);