#10218360

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

4410 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)(11,15)(12,13)(14,16)(17,21)(18,19)(20,22)(23,24)(25,26,31,32)(27,33,29,34)(28,37,38,30)(35,40,39,36);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,16,22)(12,18,17,23)(13,19,21,24)(26,32)(28,38)(33,34)(35,39);(1,3)(2,5)(4,8)(6,7)(10,16)(12,17)(15,22)(18,23)(25,27,31,29)(26,33,32,34)(28,35,38,39)(30,36,37,40);(2,6)(5,7)(9,12,14,17)(10,13,16,21)(11,18,20,23)(15,19,22,24)(25,28)(26,30)(27,35)(29,39)(31,38)(32,37)(33,36)(34,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24)(25,29,31,27)(26,34,32,33)(28,35,38,39)(30,36,37,40);(1,3)(2,7)(4,8)(5,6)(9,13)(10,17)(11,19)(12,16)(14,21)(15,23)(18,22)(20,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,38)(33,35)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^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, 10125556);