#486102

order	512
nilpotency class	4
isoclinism	isoclinic class of #486097
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

792 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,24)(20,31)(21,23)(22,27)(25,34)(26,35)(28,42)(29,37)(30,41)(32,39)(33,43)(36,40)(38,46)(44,48)(45,47);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(18,23)(21,35)(24,37)(25,38)(26,27)(29,36)(31,43)(33,46)(39,47)(40,41)(42,48)(44,45);(17,19,22,30)(18,24,27,41)(20,32,34,28)(21,36,35,29)(23,37,26,40)(25,42,31,39)(33,44,46,45)(38,48,43,47);(2,7)(3,10)(8,12)(11,13)(17,20,22,34)(18,25,27,31)(19,28,30,32)(21,33,35,46)(23,38,26,43)(24,39,41,42)(29,44,36,45)(37,47,40,48)

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=dgi,e^b=eg,f^b=fhi,d^c=di,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, 393298);