#9523443

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

4227 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9430639);