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 | #47868 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,21)(13,22)(14,16)(15,20)(18,19)(23,24)(25,26,30,33)(27,31,35,28)(29,37,39,32)(34,36,40,38);(1,2,5,7)(3,4,8,6)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,23)(22,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(10,16)(12,22)(15,17)(18,24)(25,27,30,35)(26,28,33,31)(29,34,39,40)(32,36,37,38);(9,12)(10,13)(11,18)(14,22)(15,19)(16,21)(17,23)(20,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,37)(39,40);(1,4,5,6)(2,3,7,8)(9,11)(10,17)(12,18)(13,23)(14,20)(15,16)(19,21)(22,24)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,40,37,34);(1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,19)(14,21)(15,18)(16,22)(17,24)(20,23)(25,29,30,39)(26,32,33,37)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9750919);