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 | #42555 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4342 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)(9,10,15,16)(11,18,12,13)(14,17,22,23)(19,24,20,21);(1,2)(3,7)(4,8)(5,6)(9,11,15,12)(10,13,16,18)(14,19,22,20)(17,21,23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(10,16)(11,12)(14,22)(21,24)(26,31)(29,39)(32,37)(33,40);(9,12,15,11)(10,13,16,18)(14,20,22,19)(17,21,23,24)(25,27)(26,32)(28,34)(29,33)(30,36)(31,37)(35,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13)(10,12)(11,16)(14,21)(15,18)(17,20)(19,23)(22,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,40)(32,39)(34,36);(9,14,15,22)(10,17,16,23)(11,19,12,20)(13,21,18,24)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,37,38,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7686609);