#8417142

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

4217 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)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(25,26,31,32)(27,33,37,40)(28,34,36,30)(29,35,38,39);(3,7)(6,8)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(26,32)(28,36)(29,38)(33,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,27,31,37)(26,33,32,40)(28,29,36,38)(30,39,34,35);(1,3)(2,6)(4,7)(5,8)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,28)(26,34)(27,29)(30,32)(31,36)(33,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,29)(26,35)(27,36)(28,37)(30,33)(31,38)(32,39)(34,40);(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,30,31,34)(26,28,32,36)(27,35,37,39)(29,33,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324338);