#9681724

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

4308 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9588920);