#8759303

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

4550 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)(19,24)(20,21)(22,23)(25,26)(27,29)(28,32)(30,34)(31,36)(33,38)(35,39)(37,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,23,24,21)(25,27,30,36)(26,31,34,29)(28,35,38,40)(32,37,33,39);(2,5)(6,7)(10,17)(11,12)(13,19)(21,23)(25,28,30,38)(26,32,34,33)(27,35,36,40)(29,39,31,37);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,18,17,16)(13,22,19,20)(14,23,24,21)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,34)(36,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,21,18,23)(25,29)(26,27)(28,37)(30,31)(32,40)(33,35)(34,36)(38,39);(1,3)(2,6)(4,8)(5,7)(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,22)(18,20)(25,27)(26,29)(28,35)(30,36)(31,34)(32,39)(33,37)(38,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=h,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, 8666499);