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 | |
4607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,19)(12,17)(13,18)(14,16)(15,21)(20,24)(22,23)(25,26)(27,30)(28,35)(29,32)(31,37)(33,40)(34,39)(36,38);(1,2,5,3)(4,7,8,6)(9,11,14,21)(10,15,16,19)(12,22,13,20)(17,24,18,23)(25,27)(26,30)(28,36)(29,34)(31,40)(32,39)(33,37)(35,38);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,23)(22,24)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40);(2,3)(6,7)(10,16)(15,19)(17,18)(23,24)(27,34)(28,37)(30,39)(31,35);(1,4,5,8)(2,6,3,7)(9,12,14,13)(10,17,16,18)(11,20,21,22)(15,23,19,24)(26,32)(27,34)(31,35)(33,36);(9,13,14,12)(10,18,16,17)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=g,f^2=ghi,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, 8012751);