#8159351

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

4750 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)(11,17)(12,13)(14,22)(15,16)(18,23)(19,20)(21,24)(29,37)(32,38)(34,39)(35,40);(1,2,4,6)(3,5,7,8)(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,22,24)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(10,16)(12,17)(18,22)(20,24)(26,31)(27,33)(32,38)(34,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(1,3)(2,5)(4,7)(6,8)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066547);