#9062550

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	#44303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969746);