#8933890

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841086);