#9054489

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8961685);