#9506929

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

4543 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:

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

Polycyclic presentation:

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

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, 9414125);