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 | #2535 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,7)(3,4)(9,10)(11,19)(12,23)(13,24)(14,18)(15,22)(16,21)(17,20)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,3,6,4)(2,8,7,5)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,4,6,3)(2,8,7,5)(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,17,19,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,16,19,23);(1,5,6,8)(2,4,7,3)(9,10)(11,15)(12,23)(13,17)(14,18)(16,21)(19,22)(20,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=ghi,f^2=hi,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, 8370953);