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 | #37775 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4392 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)(11,20)(12,23)(13,24)(14,19)(15,16)(17,22)(18,21)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,21)(13,22)(15,19)(16,20)(17,18)(23,24);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,22)(18,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(9,12,15,22)(10,17,16,23)(11,21,19,13)(14,18,20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,13,15,21)(10,18,16,24)(11,22,19,12)(14,17,20,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,4,5,3)(2,8,6,7)(9,14,15,20)(10,19,16,11)(12,18,22,24)(13,23,21,17)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7809914);