Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7,12,14)(5,8)(10,15)(13,16)(17,18)(19,28,31,43)(20,25)(21,26)(22,37)(23,27)(24,30,39,32)(29,44,46,40)(33,47,45,38)(34,41)(35,48)(36,42);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,33)(21,32)(22,30)(23,31)(25,40)(26,43)(27,39)(28,37)(29,35)(34,46)(36,45)(38,41)(42,44)(47,48);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,20,21,34,23,36,22,35)(18,25,26,41,27,42,37,48)(19,29,30,45,31,46,32,33)(24,38,28,40,39,47,43,44)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,e^c=eh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26535);