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 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,26,31,34)(27,29,28,32)(30,33,38,40)(35,37,36,39);(1,3,5,8)(2,6,4,7)(9,11,14,20)(10,15,16,22)(12,13,19,21)(17,18,23,24)(25,26)(27,29)(28,32)(30,33)(31,34)(35,37)(36,39)(38,40);(10,16)(15,22)(17,23)(18,24)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,13)(14,19)(15,18)(16,23)(20,21)(22,24)(25,28,31,27)(26,32,34,29)(30,35,38,36)(33,37,40,39);(1,2,5,4)(3,6,8,7)(9,13)(10,18)(11,19)(12,20)(14,21)(15,23)(16,24)(17,22)(25,29,31,32)(26,27,34,28)(30,37,38,39)(33,35,40,36);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7982454);