#8759231

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

4662 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8666427);