#486644

order	512
nilpotency class	4
isoclinism	isoclinic class of #486155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#2543	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10)(3,11,14,15)(4,8,5,9)(7,13,16,12)(17,18)(19,23)(20,30)(21,33)(22,26)(24,32)(25,35)(27,42)(28,37)(29,40)(31,41)(34,39)(36,44)(38,45)(43,47)(46,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,22,28)(18,23,26,37)(20,27,31,43)(21,29,35,45)(24,36,39,48)(25,40,33,38)(30,47,41,42)(32,44,34,46);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,41)(26,39)(28,43)(29,46)(30,33)(32,35)(37,48)(38,47)(40,42)(44,45);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(19,28)(20,31)(23,37)(24,39)(29,45)(30,41)(32,34)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393840);