Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,29,40)(20,25,31,37)(22,34,33,26)(24,35,36,30)(28,41,42,48)(32,45,44,39)(38,46,47,43);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(20,31)(24,36)(26,34)(27,40)(28,42)(32,44)(43,46);(11,15)(13,16)(17,19,21,29)(18,24,23,36)(20,28,31,42)(22,35,33,30)(25,38,37,47)(26,40,34,27)(32,46,44,43)(39,48,45,41);(11,15)(13,16)(17,20,21,31)(18,25,23,37)(19,28,29,42)(22,32,33,44)(24,38,36,47)(26,39,34,45)(27,41,40,48)(30,43,35,46);(10,14)(13,16)(20,31)(25,37)(28,42)(32,44)(38,47)(39,45)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,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, 6377707);