#219666

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

457 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126862);