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 | |
4479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,14)(12,19)(13,21)(15,16)(17,22)(18,23)(20,24)(25,26,30,32)(27,33,34,28)(29,31,39,40)(35,38,37,36);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,14,16,19)(13,18,22,20)(17,24,21,23)(25,27)(26,28)(29,37)(30,34)(31,38)(32,33)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(10,16)(11,12)(13,22)(23,24)(27,34)(28,33)(35,37)(36,38);(1,4,5,3)(2,8,6,7)(9,12,15,11)(10,14,16,19)(13,20,22,18)(17,24,21,23)(25,26)(27,28)(29,31)(30,32)(33,34)(35,36)(37,38)(39,40);(9,13)(10,17)(11,18)(12,20)(14,24)(15,22)(16,21)(19,23)(25,28)(26,27)(29,38)(30,33)(31,37)(32,34)(35,40)(36,39);(9,14)(10,11)(12,16)(13,23)(15,19)(17,20)(18,21)(22,24)(25,29,30,39)(26,31,32,40)(27,35,34,37)(28,36,33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894494);