#9034295

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=gi,e^2=g,f^2=i,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, 8941491);