#8395290

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

4554 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8302486);