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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,20)(12,14)(13,23)(15,19)(16,24)(17,22)(18,21)(25,26)(27,32)(28,33)(29,39)(30,35)(31,36)(34,38)(37,40);(2,6)(3,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,40)(38,39);(1,3)(2,4)(5,8)(6,7)(25,27,31,40)(26,32,36,37)(28,38,35,29)(30,39,33,34);(1,3)(2,4)(5,8)(6,7)(9,12,15,22)(10,17,19,14)(11,21,24,13)(16,18,20,23)(25,28)(26,33)(27,38)(29,40)(30,36)(31,35)(32,34)(37,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,21)(10,18,19,23)(11,22,24,12)(14,16,17,20)(25,29,31,38)(26,34,36,39)(27,28,40,35)(30,37,33,32);(9,14,15,17)(10,12,19,22)(11,23,24,18)(13,16,21,20)(25,30,31,33)(26,35,36,28)(27,39,40,34)(29,37,38,32)Polycyclic presentation:
2^9|c^2=hi,d^2=gh,e^2=gi,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, 8371436);