#219684

order	512
nilpotency class	3
isoclinism	isoclinic class of #219624
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

453 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)(3,8,15,23,13,22,14,29)(4,16,18,24,19,30,17,9)(12,25,28,31,26,32,27,21)(33,34,38,44,39,45,50,43)(35,46,48,40,49,36,41,52)(37,53,54,42,55,58,64,59)(47,61,62,60,63,56,57,51);(1,3,6,15,7,13,5,14)(2,8,10,23,11,22,20,29)(4,12,18,28,19,26,17,27)(9,21,16,25,24,31,30,32)(33,35,39,49)(34,40)(36,44)(37,47,55,63)(38,48,50,41)(42,56)(43,46)(45,52)(51,58)(53,60)(54,62,64,57)(59,61);(1,4,5,17,7,19,6,18)(2,9,10,16,11,24,20,30)(3,12,14,27,13,26,15,28)(8,21,23,25,22,31,29,32)(33,36,50,52,39,46,38,40)(34,41,43,35,45,48,44,49)(37,51,64,61,55,60,54,56)(42,57,53,47,59,62,58,63);(3,13)(8,22)(12,26)(14,15)(21,31)(23,29)(25,32)(27,28)(33,37,38,54,39,55,50,64)(34,42,43,53,45,59,44,58)(35,47,48,62,49,63,41,57)(36,51,40,56,46,60,52,61)

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=dhi,f^b=fh,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, 126880);