Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1409 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)(11,15)(14,16)(17,18,24,30,23,29,22,28)(19,31,35,42,34,25,33,43)(20,27,37,45,21,38,39,26)(32,41,47,40,36,46,48,44);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,36)(21,32)(22,33)(23,34)(24,35)(26,44)(27,41)(28,42)(29,31)(30,43)(37,48)(38,46)(39,47)(40,45);(10,13)(14,16)(17,20,23,21)(18,26,29,45)(19,32,34,36)(22,37,24,39)(25,40,31,44)(27,30,38,28)(33,47,35,48)(41,42,46,43);(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,38)(19,32,34,36)(22,37,24,39)(25,41,31,46)(26,30,45,28)(33,48,35,47)(40,42,44,43);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,28,29,30)(19,33,34,35)(20,37,21,39)(25,42,31,43)(26,27,45,38)(32,47,36,48)(40,46,44,41)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343004);