Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(12,20)(15,23)(19,24)(21,22)(26,30)(28,35)(31,36)(34,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,21)(20,23)(22,24)(27,33)(31,36)(34,39)(38,40);(9,11)(10,13)(12,19)(14,17)(15,22)(16,18)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,20)(10,15,18,23)(11,19,17,24)(13,22,16,21)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,16)(12,19)(13,18)(14,17)(15,21)(20,24)(22,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,17,18,11)(12,21,20,22)(15,19,23,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8595567);