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 | #45994 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,14,18)(11,17,13,15)(12,16,21,24)(19,23,20,22)(25,26,30,31)(27,35,38,32)(28,29,34,33)(36,39,40,37);(1,2,5,3)(4,6,8,7)(9,11,14,13)(10,15,18,17)(12,19,21,20)(16,22,24,23)(26,31)(28,34)(32,35)(36,40);(2,3)(6,7)(9,12,14,21)(10,16,18,24)(11,19,13,20)(15,22,17,23)(25,27,30,38)(26,32,31,35)(28,36,34,40)(29,37,33,39);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,28,30,34)(26,33,31,29)(27,36,38,40)(32,39,35,37);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24);(1,4)(2,6)(3,7)(5,8)(9,13,14,11)(10,17,18,15)(12,20,21,19)(16,23,24,22)(25,29,30,33)(26,34,31,28)(27,37,38,39)(32,40,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8001934);