Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,15,17)(11,18,21,16)(12,13,22,14)(19,23,24,20)(25,26)(27,36)(28,40)(29,35)(30,32)(31,44)(33,39)(34,42)(37,48)(38,53)(41,50)(43,49)(45,46)(47,54)(51,52)(55,56);(1,3,4,8)(2,5,6,7)(9,11,15,21)(10,16,17,18)(12,19,22,24)(13,20,14,23)(26,32)(27,37)(34,40)(35,44)(38,52)(39,45)(49,50)(54,55);(3,8)(5,7)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,23)(21,24)(25,27,30,37)(26,33,32,46)(28,38,42,52)(29,39,31,45)(34,47,40,56)(35,48,44,36)(41,54,43,55)(49,51,50,53);(2,6)(5,7)(9,13,15,14)(10,12,17,22)(11,20,21,23)(16,19,18,24)(25,28)(26,34)(27,38)(29,43)(30,42)(31,41)(32,40)(33,47)(35,50)(36,51)(37,52)(39,55)(44,49)(45,54)(46,56)(48,53);(9,14,15,13)(10,12,17,22)(11,20,21,23)(16,24,18,19)(25,29,30,31)(26,35,32,44)(27,39,37,45)(28,41,42,43)(33,48,46,36)(34,49,40,50)(38,54,52,55)(47,51,56,53)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6893964);