#9523781

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
prime radical	#42068	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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)(12,16)(13,24)(14,18)(17,21)(19,20)(22,23)(25,26,31,32)(27,28,29,33)(30,38,39,34)(35,40,37,36);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24)(26,32)(27,29)(34,38)(35,37);(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,19,20,15)(12,22,23,16)(13,24,21,17)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(3,8)(5,7)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,13)(14,23)(15,24)(17,19)(18,22)(20,21)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,12)(14,21)(15,22)(16,19)(18,24)(20,23)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,38)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=ghi,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, 9430977);