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 | |
4585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(12,20)(21,23)(25,26)(27,28)(29,37)(30,33)(31,36)(32,38)(34,40)(35,39);(1,2,5,6)(3,7,8,4)(9,10,14,15)(11,13,18,16)(12,17,20,22)(19,21,24,23)(25,27,30,31)(26,28,33,36)(29,34,38,39)(32,35,37,40);(2,6)(4,7)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,23,22,21)(25,28,30,36)(26,31,33,27)(29,35,38,40)(32,39,37,34);(1,3)(2,4)(5,8)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24);(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,37)(36,40);(1,4)(2,3)(5,7)(6,8)(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,26)(27,28)(29,32)(30,33)(31,36)(34,35)(37,38)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8088329);