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 |
| prime radical | #47851 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(26,31)(29,33)(32,38)(35,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,18)(10,15,16,13)(12,19,20,24)(17,22,23,21)(25,26)(27,32)(28,29)(30,31)(33,37)(34,35)(36,38)(39,40);(10,16)(11,18)(12,20)(21,22)(26,31)(28,37)(32,38)(34,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,33,39);(9,11)(10,13)(12,19)(14,18)(15,16)(17,21)(20,24)(22,23)(25,28)(26,33)(27,34)(29,31)(30,37)(32,39)(35,38)(36,40);(1,4,5,3)(2,8,6,7)(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)(25,29,30,33)(26,28,31,37)(27,35,36,39)(32,34,38,40)Polycyclic presentation:
2^9|b^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622951);