#484757

order	512
nilpotency class	4
isoclinism	isoclinic class of #484606
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×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391953);