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 | #39491 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,19)(12,18)(13,15)(14,17)(16,20)(21,23)(22,24)(25,26)(27,29)(28,32)(30,33)(31,35)(34,39)(36,37)(38,40);(1,2,5,4)(3,7,8,6)(9,11,16,14)(10,17,20,19)(12,21,13,22)(15,23,18,24)(25,27,30,35)(26,29,33,31)(28,36,38,34)(32,37,40,39);(1,3,5,8)(2,6,4,7)(9,12,16,13)(10,18,20,15)(11,21,14,22)(17,24,19,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,38)(39,40);(1,4,5,2)(3,7,8,6)(9,13,16,12)(10,18,20,15)(11,22,14,21)(17,24,19,23)(25,28,30,38)(26,32,33,40)(27,34,35,36)(29,39,31,37);(9,14,16,11)(10,19,20,17)(12,22,13,21)(15,24,18,23);(9,15,16,18)(10,13,20,12)(11,23,14,24)(17,22,19,21)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,36,40,34)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366287);