Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1606 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,33)(21,27)(22,24)(23,37)(25,32)(26,35)(29,43)(30,45)(31,38)(34,41)(36,40)(39,47)(42,44)(46,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(19,29)(26,33)(27,37)(30,44)(31,32)(40,41)(46,47);(10,14)(11,15)(17,19,23,32,22,29,21,31)(18,25,27,28,24,38,37,43)(20,30,34,46,35,44,36,47)(26,39,41,42,33,48,40,45);(2,6)(5,7)(17,20,22,35)(18,26,24,33)(19,30,29,44)(21,36,23,34)(25,39,38,48)(27,41,37,40)(28,42,43,45)(31,47,32,46);(3,8)(5,7)(11,15)(13,16)(17,21,22,23)(18,27,24,37)(19,31,29,32)(20,34,35,36)(25,28,38,43)(26,40,33,41)(30,46,44,47)(39,45,48,42)Polycyclic presentation:
2^9|b^2=fi,c^2=ghi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669423);