Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #31068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,29,37)(20,30,33,25)(21,26,35,36)(27,41,38,31)(28,34,40,47)(32,43,46,39)(42,48,45,44);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,22,29)(18,24,23,37)(20,31,33,41)(21,34,35,47)(25,38,30,27)(26,40,36,28)(32,45,46,42)(39,48,43,44);(10,13)(14,16)(17,20,18,25,22,33,23,30)(19,27,37,41,29,38,24,31)(21,32,26,39,35,46,36,43)(28,42,47,48,40,45,34,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(21,35)(26,36)(28,40)(32,46)(34,47)(39,43)(42,45)(44,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,36,22,35,18,26)(19,28,24,34,29,40,37,47)(20,32,30,43,33,46,25,39)(27,42,31,44,38,45,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=fgi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6747852);