#219675

order	512
nilpotency class	3
isoclinism	isoclinic class of #219600
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	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

466 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ehi,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=di,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, 126871);