#9039275

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

4732 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)(3,5)(4,8)(6,7)(9,10,14,15)(11,16,20,23)(12,17,21,13)(18,19,22,24)(27,33)(31,34)(35,39)(38,40);(2,7)(3,8)(10,15)(12,21)(16,23)(18,22)(25,26,29,30)(27,31,33,34)(28,32,36,37)(35,38,39,40);(1,3,6,8)(2,4,7,5)(12,21)(13,17)(18,22)(19,24)(26,30)(28,36)(31,34)(35,39);(1,4,6,5)(2,3,7,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,23)(17,24)(18,21)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,2)(3,4)(5,8)(6,7)(9,12)(10,17)(11,18)(13,15)(14,21)(16,19)(20,22)(23,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,5,6,4)(2,8,7,3)(9,13,14,17)(10,12,15,21)(11,19,20,24)(16,22,23,18)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946471);