#8807297

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	#44315	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(11,13)(12,17)(14,15)(16,19)(18,21)(20,24)(22,23)(26,31)(27,33)(29,37)(35,40);(1,2)(3,7)(4,5)(6,8)(10,15)(13,16)(17,23)(21,24)(25,26,30,31)(27,28,33,36)(29,38,37,32)(34,40,39,35);(2,5)(6,7)(9,11,14,19)(10,13,15,16)(12,20,22,18)(17,24,23,21)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,18)(13,15)(14,19)(17,24)(20,22)(21,23)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,15,23)(11,18,19,20)(13,21,16,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,3,4,8)(2,6,5,7)(9,13,14,16)(10,11,15,19)(12,21,22,24)(17,18,23,20)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714493);