#9034386

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

4826 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)(9,10,13,15)(11,14,17,23)(12,19,21,16)(18,24,20,22)(25,26,31,29)(27,30,35,32)(28,33,37,38)(34,36,40,39);(3,7)(5,8)(9,11)(10,14)(12,18)(13,17)(15,23)(16,22)(19,24)(20,21)(25,27,31,35)(26,32,29,30)(28,34,37,40)(33,39,38,36);(1,2)(3,5)(4,6)(7,8)(10,15)(11,17)(16,19)(18,20)(26,29)(30,32)(33,38)(36,39);(9,12)(10,16)(11,18)(13,21)(14,22)(15,19)(17,20)(23,24)(25,28)(26,33)(27,34)(29,38)(30,36)(31,37)(32,39)(35,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,20)(13,17)(15,23)(16,24)(18,21)(19,22)(25,29,31,26)(27,30,35,32)(28,33,37,38)(34,39,40,36);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,23)(18,22)(19,21)(20,24)(25,30)(26,27)(28,36)(29,35)(31,32)(33,34)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^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, 8941582);