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 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,13)(12,16)(14,15)(17,22)(18,21)(19,24)(20,23)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(2,6)(3,7)(10,15)(11,17)(16,23)(18,24)(25,27,30,35)(26,31,28,32)(29,34,38,37)(33,39,36,40);(1,3)(2,4)(5,7)(6,8)(10,15)(11,17)(16,23)(18,24);(1,4)(2,3)(5,8)(6,7)(9,11,14,17)(10,13,15,22)(12,18,20,24)(16,21,23,19)(25,28,30,26)(27,31,35,32)(29,36,38,33)(34,39,37,40);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854184);