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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,15)(13,23)(16,22)(17,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,39)(37,38);(2,5)(3,7)(11,19)(12,21)(16,22)(17,24)(26,31)(32,38)(33,34)(39,40);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,16,19,22)(12,17,21,24)(13,18,23,20)(25,27,30,37)(26,32,31,38)(28,35,29,36)(33,39,34,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,17)(14,19)(15,22)(18,21)(23,24)(25,28,30,29)(26,33,31,34)(27,35,37,36)(32,39,38,40);(9,12,14,21)(10,17,15,24)(11,20,19,18)(13,22,23,16)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,13)(10,18)(11,17)(12,22)(14,23)(15,20)(16,21)(19,24)(25,29,30,28)(26,34,31,33)(27,36,37,35)(32,40,38,39)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000499);