#10055586

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

4773 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9962782);