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 |
| prime radical | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4256 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)(10,16)(12,22)(17,19)(20,23)(26,31)(27,35)(33,34)(36,37);(1,2)(3,7)(4,8)(5,6)(9,10)(11,19)(12,14)(13,17)(15,16)(18,22)(20,24)(21,23)(26,31)(32,38)(33,34)(39,40);(9,11,15,13)(10,17,16,19)(12,20,22,23)(14,21,18,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(9,12)(10,18)(11,20)(13,23)(14,16)(15,22)(17,24)(19,21)(25,27,30,35)(26,32,31,38)(28,36,29,37)(33,39,34,40);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,17,16,19)(12,23,22,20)(14,21,18,24)(25,28,30,29)(26,33,31,34)(27,36,35,37)(32,39,38,40);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,12,16,22)(11,21,13,24)(17,20,19,23)(25,29,30,28)(26,34,31,33)(27,37,35,36)(32,40,38,39)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566175);