#9523445

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

4389 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 9430641);