#9779476

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

4467 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)(11,18)(12,13)(14,16)(15,22)(17,19)(20,23)(21,24)(25,26,31,34)(27,30,33,28)(29,32,38,40)(35,39,37,36);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,23)(19,24)(25,27,31,33)(26,28,34,30)(29,37,38,35)(32,39,40,36);(1,2)(3,6)(4,5)(7,8)(10,16)(12,19)(15,18)(20,24)(25,28,31,30)(26,27,34,33)(29,36,38,39)(32,35,40,37);(9,12,14,19)(10,17,16,13)(11,20,22,24)(15,23,18,21)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,18)(12,20)(13,23)(14,22)(15,16)(17,21)(19,24)(25,29)(26,32)(27,35)(28,36)(30,39)(31,38)(33,37)(34,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,17)(10,19,16,12)(11,21,22,23)(15,24,18,20)(25,30,31,28)(26,33,34,27)(29,36,38,39)(32,35,40,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9686672);