Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4568 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)(3,4)(9,10,15,12)(11,19,22,16)(13,18,14,17)(20,23,21,24)(25,26,30,31)(27,35,37,32)(28,34,29,33)(36,39,38,40);(1,2)(3,8)(4,7)(5,6)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,23)(18,24)(26,31)(28,29)(32,35)(36,38);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,19,22,16)(13,17,14,18)(20,23,21,24)(25,27,30,37)(26,32,31,35)(28,38,29,36)(33,40,34,39);(1,3,5,4)(2,7,6,8)(9,13,15,14)(10,17,12,18)(11,20,22,21)(16,23,19,24)(25,28,30,29)(26,33,31,34)(27,36,37,38)(32,39,35,40);(1,4,5,3)(2,7,6,8)(9,13,15,14)(10,18,12,17)(11,21,22,20)(16,23,19,24)(25,29,30,28)(26,34,31,33)(27,36,37,38)(32,39,35,40);(9,14,15,13)(10,17,12,18)(11,21,22,20)(16,23,19,24)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007543);