#487349

order	512
nilpotency class	4
isoclinism	isoclinic class of #486803
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
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

822 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,40)(29,41)(31,37)(32,38)(33,46)(34,39)(35,43)(36,44)(42,48)(45,47);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,35)(25,27)(26,36)(31,43)(32,42)(33,45)(34,44)(37,40)(38,47)(39,41)(46,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,20,22,32)(18,24,26,38)(19,28,29,42)(21,33,34,31)(23,35,36,47)(25,37,39,46)(27,40,41,48)(30,45,44,43);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(19,29)(23,36)(27,41)(28,42)(30,44)(35,47)(40,48)(43,45)

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,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, 394545);