#487646

order	512
nilpotency class	4
isoclinism	isoclinic class of #487643
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	#18356	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

783 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18)(19,28)(20,25)(21,33)(22,36)(23,27)(24,30)(26,38)(29,44)(31,43)(32,48)(34,45)(35,42)(37,41)(39,46)(40,47);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,29)(21,34)(22,37)(23,31)(25,39)(26,43)(27,41)(28,36)(30,38)(32,47)(33,40)(35,46)(42,45)(44,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,23,21)(18,25,27,33)(19,29,31,34)(22,35,38,32)(24,39,41,40)(26,42,36,48)(28,44,43,45)(30,47,37,46);(2,8)(3,10)(6,9)(11,16)(17,21,23,20)(18,25,27,33)(19,29,31,34)(22,35,38,32)(24,40,41,39)(26,42,36,48)(28,45,43,44)(30,46,37,47)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394842);