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 | #59812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,19)(12,13)(14,16)(15,17)(18,24)(20,21)(22,23);(1,2)(3,6)(4,7)(5,8)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,23,24,21)(25,26)(27,32)(28,29)(30,31)(33,38)(34,40)(35,37)(36,39);(1,3,5,4)(2,6,8,7)(10,17)(11,14)(12,18)(21,23)(26,31)(29,33)(32,39)(35,40);(1,3,5,4)(2,6,8,7)(9,12)(10,13)(11,20)(14,22)(15,18)(16,23)(17,24)(19,21)(25,27)(26,32)(28,37)(29,35)(30,36)(31,39)(33,40)(34,38);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,20)(19,22)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(9,14,15,11)(10,16,17,19)(12,22,18,20)(13,23,24,21)(25,29,30,33)(26,28,31,38)(27,35,36,40)(32,37,39,34)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7706742);