#9161719

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

4454 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068915);