Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,20,25)(21,26,36,44)(22,27,34,42)(29,37,32,40)(30,38,31,39)(33,41,46,48)(35,43,45,47);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,28,25)(21,33,36,46)(22,30,34,31)(26,41,44,48)(27,38,42,39)(29,45,32,35)(37,47,40,43);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,19)(18,25,28,24)(21,29,36,32)(22,30,34,31)(26,37,44,40)(27,38,42,39)(33,45,46,35)(41,47,48,43);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,34,45,23,36,22,35)(18,26,42,47,28,44,27,43)(19,29,30,46,20,32,31,33)(24,37,38,48,25,40,39,41);(2,6)(7,8)(17,22,23,34)(18,27,28,42)(19,30,20,31)(21,35,36,45)(24,38,25,39)(26,43,44,47)(29,46,32,33)(37,48,40,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517240);