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 | #36374 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4435 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)(9,10,15,12)(11,18,17,14)(13,20,21,16)(19,22,23,24)(25,26,31,28)(27,35,36,32)(29,33,39,38)(30,37,40,34);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,24)(21,23)(25,27,31,36)(26,32,28,35)(29,37,39,34)(30,38,40,33);(10,12)(11,17)(13,21)(22,24)(27,36)(29,39)(32,35)(33,38);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,17,14)(13,20,21,16)(19,24,23,22)(25,28,31,26)(27,35,36,32)(29,38,39,33)(30,37,40,34);(1,4,5,3)(2,8,6,7)(9,13,15,21)(10,16,12,20)(11,19,17,23)(14,22,18,24)(25,29,31,39)(26,33,28,38)(27,37,36,34)(30,35,40,32);(9,14,15,18)(10,17,12,11)(13,22,21,24)(16,23,20,19)(25,30,31,40)(26,34,28,37)(27,38,36,33)(29,35,39,32)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619329);