#487476

order	512
nilpotency class	4
isoclinism	isoclinic class of #486819
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	#3459	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

729 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)(4,7)(5,9)(6,14)(8,12)(11,15)(13,16)(17,18)(19,27)(20,24)(21,34)(22,26)(23,29)(25,37)(28,44)(30,43)(31,47)(32,41)(33,48)(35,39)(36,42)(38,45)(40,46);(1,3)(2,8)(4,11)(5,12)(6,15)(7,13)(9,16)(10,14)(17,19,22,30)(18,23,26,39)(20,31,32,28)(21,35,37,29)(24,40,41,38)(25,43,34,27)(33,45,36,46)(42,44,48,47);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,20,22,32)(18,24,26,41)(19,28,30,31)(21,36,37,33)(23,38,39,40)(25,42,34,48)(27,44,43,47)(29,46,35,45);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(20,32)(24,41)(28,31)(33,36)(38,40)(42,48)(44,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394672);