#9712777

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

4316 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 9619973);