#484607

order	512
nilpotency class	4
isoclinism	isoclinic class of #484606
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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	#18292	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,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, 391803);