#9515551

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

4379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=ghi,f^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, 9422747);