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 | |
4450 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)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,35)(28,38)(29,36)(30,34)(31,37)(32,39)(33,40);(1,3,5,8)(2,6,4,7)(25,27,30,37)(26,31,34,35)(28,33,39,36)(29,32,40,38);(1,2,5,4)(3,6,8,7)(10,14)(15,17)(16,21)(22,23)(25,28,30,39)(26,32,34,38)(27,36,37,33)(29,35,40,31);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,32)(27,33)(29,35)(30,39)(31,40)(34,38)(36,37);(1,2,5,4)(3,6,8,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29)(26,33)(27,32)(28,35)(30,40)(31,39)(34,36)(37,38);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,36)(30,34)(33,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7805044);