#7714477

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

4311 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621673);