#9138439

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

4631 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)(11,16)(12,20)(13,17)(14,18)(15,21)(19,23)(22,24)(25,26,30,32)(27,34,37,31)(28,36,38,33)(29,39,40,35);(1,2)(3,4)(5,7)(6,8)(9,11)(10,13)(12,19)(14,17)(15,23)(16,18)(20,24)(21,22)(25,27,30,37)(26,31,32,34)(28,35,38,39)(29,36,40,33);(1,3,5,8)(2,6,7,4)(12,21)(15,20)(19,22)(23,24)(26,32)(28,38)(31,34)(35,39);(3,8)(4,6)(9,12,14,21)(10,15,18,20)(11,19,17,22)(13,23,16,24)(25,28,30,38)(26,33,32,36)(27,35,37,39)(29,34,40,31);(1,4)(2,3)(5,6)(7,8)(9,11)(10,16)(12,22)(13,18)(14,17)(15,23)(19,21)(20,24)(25,27)(26,34)(28,39)(29,33)(30,37)(31,32)(35,38)(36,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,16)(10,17,18,11)(12,23,21,24)(15,22,20,19)(25,29)(26,35)(27,36)(28,34)(30,40)(31,38)(32,39)(33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9045635);