#10040832

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

4480 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)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26,31,34)(27,32,30,40)(28,33,38,29)(35,36,37,39);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,14)(13,19)(16,23)(17,18)(20,22)(21,24)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,15,12,18)(13,16,22,21)(19,23,20,24)(25,28,31,38)(26,29,34,33)(27,35,30,37)(32,39,40,36);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(9,12,14,11)(10,15,17,18)(13,19,22,20)(16,24,21,23)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,e^2=h,f^2=h,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, 9948028);