#8418131

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

4397 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=gh,f^2=gi,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, 8325327);