Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,31,36)(27,32,40,49)(28,34,43,51)(29,35,30,45)(33,44,52,42)(37,47,41,53)(38,48,39,56)(46,55,50,54);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,27,31,40)(26,32,36,49)(28,41,43,37)(29,38,30,39)(33,50,52,46)(34,53,51,47)(35,48,45,56)(42,55,44,54);(1,3)(2,5)(4,8)(6,7)(25,28,31,43)(26,33,36,52)(27,37,40,41)(29,42,30,44)(32,46,49,50)(34,35,51,45)(38,54,39,55)(47,48,53,56);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,34)(27,37)(29,44)(30,42)(31,43)(32,47)(33,45)(35,52)(36,51)(38,55)(39,54)(40,41)(46,56)(48,50)(49,53);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,35,36,45)(27,38,40,39)(28,42,43,44)(32,48,49,56)(33,51,52,34)(37,54,41,55)(46,53,50,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7213805);