#9032486

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

4548 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8939682);