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 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(1,2)(3,7)(4,8)(5,6)(9,11,14,20)(10,15,18,19)(12,21,23,13)(16,17,22,24);(11,20)(13,21)(15,19)(17,24)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(9,12)(10,16)(11,21)(13,20)(14,23)(15,17)(18,22)(19,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,12,20,23)(15,22,19,16)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7559143);