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 | |
2216 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)(17,18,22,24)(19,28,31,40)(20,26,33,38)(21,27,23,35)(25,36,37,30)(29,43,45,48)(32,41,34,42)(39,44,47,46);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(25,37)(26,38)(27,35)(28,40)(39,47)(41,42)(43,48);(11,15)(13,16)(17,19)(18,25)(20,29)(21,30)(22,31)(23,36)(24,37)(26,39)(27,40)(28,35)(32,44)(33,45)(34,46)(38,47)(41,43)(42,48);(11,15)(13,16)(17,20,22,33)(18,26,24,38)(19,29,31,45)(21,34,23,32)(25,39,37,47)(27,42,35,41)(28,43,40,48)(30,46,36,44);(10,14)(13,16)(17,21,22,23)(18,27,24,35)(19,30,31,36)(20,32,33,34)(25,40,37,28)(26,41,38,42)(29,44,45,46)(39,43,47,48)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6377695);