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 | |
4444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26)(27,34)(28,36)(29,32)(30,33)(31,38)(35,37)(39,40);(2,3)(6,7)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,30,32)(26,29,33,34)(28,35,38,39)(31,37,36,40);(1,2,5,3)(4,6,8,7)(10,15)(11,18)(16,23)(19,20);(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,39)(33,36)(34,40);(1,3,5,2)(4,6,8,7)(9,12,14,22)(10,16,15,23)(11,19,18,20)(13,21,17,24)(25,29,30,34)(26,32,33,27)(28,37,38,40)(31,39,36,35);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9755849);