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 | |
4677 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)(9,10,15,17)(11,16,21,14)(12,18,20,24)(13,19,23,22)(25,26,30,32)(27,31,35,38)(28,33,37,39)(29,34,36,40);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,17,14)(12,13,20,23)(18,19,24,22)(25,27,30,35)(26,31,32,38)(28,36,37,29)(33,40,39,34);(1,2,5,4)(3,6,8,7)(10,17)(14,16)(18,24)(19,22)(26,32)(27,35)(29,36)(33,39);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,13)(14,22)(15,20)(16,19)(17,24)(21,23)(25,28,30,37)(26,33,32,39)(27,29,35,36)(31,34,38,40);(9,13)(10,19)(11,20)(12,21)(14,18)(15,23)(16,24)(17,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(9,14)(10,11)(12,22)(13,18)(15,16)(17,21)(19,20)(23,24)(25,29)(26,34)(27,28)(30,36)(31,33)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7995292);