#9050391

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

4375 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8957587);