#9029074

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

4287 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=g,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, 8936270);