#8284188

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

4738 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8191384);