#8374220

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

4275 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8281416);