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 | #59406 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4522 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)(11,17)(12,15)(13,18)(14,16)(19,24)(20,22)(21,23)(25,26)(27,29)(28,36)(30,32)(31,34)(33,37)(35,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,16,17)(13,19,21,20)(18,22,23,24)(25,27,30,34)(26,31,32,29)(28,35,37,40)(33,39,36,38);(10,16)(15,17)(18,23)(22,24)(26,32)(27,34)(28,37)(38,39);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,16,15)(13,20,21,19)(18,24,23,22)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,29,30,31)(26,34,32,27)(28,38,37,39)(33,40,36,35)Polycyclic presentation:
2^9|b^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618115);