#480373

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1467
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,e^2=i,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fgi,f^c=fi,f^e=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, 387569);