#8256565

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

4424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8163761);