#486206

order	512
nilpotency class	4
isoclinism	isoclinic class of #486147
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×C₂²⋊C₄	C₂⁴
prime radical	#1595	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

750 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,31)(21,35)(22,26)(23,30)(24,34)(25,37)(28,42)(29,44)(32,43)(33,41)(36,38)(39,47)(40,45)(46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,32)(21,36)(22,28)(24,40)(25,42)(26,38)(27,35)(29,34)(30,37)(31,39)(33,45)(41,43)(44,46)(47,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,28)(21,37)(23,38)(27,42)(29,32)(33,48)(39,40)(43,44);(2,9)(3,12)(8,13)(11,14)(17,20,22,34)(18,24,26,31)(19,29,28,32)(21,33,37,48)(23,39,38,40)(25,41,35,46)(27,43,42,44)(30,45,36,47)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,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, 393402);