Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1728 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)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,27,30,24)(20,31,34,25)(21,35,38,26)(28,39,32,41)(29,40,46,36)(33,42,37,43)(44,48,45,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,23,27)(20,32,34,28)(21,36,38,40)(25,41,31,39)(26,29,35,46)(33,47,37,48)(42,45,43,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,34)(18,25,23,31)(19,28,30,32)(21,37,38,33)(24,39,27,41)(26,43,35,42)(29,45,46,44)(36,47,40,48);(10,14)(13,15)(17,21,22,38)(18,26,23,35)(19,29,30,46)(20,33,34,37)(24,40,27,36)(25,42,31,43)(28,44,32,45)(39,47,41,48);(2,5)(6,8)(10,14)(13,15)(20,34)(25,31)(28,32)(33,37)(39,41)(42,43)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7439269);