#10219579

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10126775);