#478283

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi

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, 385479);