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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1313 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,23,28)(19,24,31,38)(20,25,35,42)(21,26,33,40)(22,27,30,37)(29,36,45,48)(32,39,44,47)(34,41,43,46);(9,10)(11,15)(12,16)(13,14)(17,19,23,31)(18,24,28,38)(20,32,35,44)(21,30,33,22)(25,39,42,47)(26,37,40,27)(29,43,45,34)(36,46,48,41);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,33,43,23,35,21,34)(18,25,40,46,28,42,26,41)(19,29,30,44,31,45,22,32)(24,36,37,47,38,48,27,39);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,23,33)(18,26,28,40)(19,30,31,22)(20,34,35,43)(24,37,38,27)(25,41,42,46)(29,44,45,32)(36,47,48,39);(10,14)(15,16)(17,22,21,31,23,30,33,19)(18,27,26,38,28,37,40,24)(20,32,34,45,35,44,43,29)(25,39,41,48,42,47,46,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=gi,e^2=h,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, 6729519);