Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 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)(17,18,23,24,22,27,19,28)(20,31,29,40,34,37,32,25)(21,26,36,38,35,41,30,42)(33,44,43,48,46,47,45,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,35,36)(25,31,40,37)(26,38,41,42)(33,45,46,43)(39,44,48,47);(10,14)(15,16)(17,20,22,34)(18,25,27,40)(19,29,23,32)(21,33,35,46)(24,37,28,31)(26,39,41,48)(30,43,36,45)(38,47,42,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,35)(18,26,27,41)(19,30,23,36)(20,33,34,46)(24,38,28,42)(25,39,40,48)(29,43,32,45)(31,44,37,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(21,35)(26,41)(30,36)(33,46)(38,42)(39,48)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6774834);