#8393387

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

4430 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)(3,6)(4,5)(7,8)(9,10,15,16)(11,19,22,14)(12,23,21,17)(13,24,20,18)(25,26,30,31)(27,29,33,39)(28,35,38,32)(34,40,36,37);(1,2,4,5)(3,6,7,8)(9,11,15,22)(10,14,16,19)(12,20,21,13)(17,18,23,24)(25,26,30,31)(27,29,33,39)(28,35,38,32)(34,40,36,37);(3,7)(6,8)(10,16)(14,19)(17,23)(18,24)(26,31)(29,39)(32,35)(37,40);(2,5)(6,8)(9,12)(10,17)(11,20)(13,22)(14,18)(15,21)(16,23)(19,24)(25,27,30,33)(26,29,31,39)(28,36,38,34)(32,40,35,37);(1,2)(3,6)(4,5)(7,8)(9,13,15,20)(10,18,16,24)(11,21,22,12)(14,17,19,23)(25,28,30,38)(26,32,31,35)(27,34,33,36)(29,37,39,40);(1,3,4,7)(2,6,5,8)(9,14,15,19)(10,11,16,22)(12,18,21,24)(13,17,20,23)(25,29)(26,33)(27,31)(28,37)(30,39)(32,36)(34,35)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=i,e^2=h,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, 8300583);