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 | |
4305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,17,21,15)(12,22,18,13)(19,20,24,23)(25,26)(27,34)(28,36)(29,32)(30,33)(31,38)(35,37)(39,40);(2,3)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,17)(18,24)(22,23)(25,27,30,32)(26,29,33,34)(28,35,38,39)(31,37,36,40);(1,2,5,3)(4,6,8,7)(10,16)(12,18)(15,17)(19,24);(9,12,14,18)(10,13,16,22)(11,19,21,24)(15,20,17,23)(25,28)(26,31)(27,35)(29,37)(30,38)(32,39)(33,36)(34,40);(1,3,5,2)(4,6,8,7)(9,11)(10,17)(12,19)(13,23)(14,21)(15,16)(18,24)(20,22)(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)(10,18)(11,20)(12,16)(14,22)(15,24)(17,19)(21,23)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=h,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, 9756067);