Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1318 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)(27,35)(28,39)(29,33)(30,43)(31,37)(32,41)(34,42)(36,51)(38,46)(40,48)(44,47)(45,55)(49,53)(50,54)(52,56);(1,3,4,8)(2,5,6,7)(9,11,15,21)(10,16,17,18)(12,19,22,24)(13,20,14,23)(25,27,29,37)(26,31,33,35)(28,36,40,52)(30,38,42,54)(32,45,47,49)(34,46,43,50)(39,56,48,51)(41,53,44,55);(3,8)(5,7)(9,12)(10,13)(11,19)(14,17)(15,22)(16,20)(18,23)(21,24)(25,28)(26,32)(27,36)(29,40)(30,44)(31,45)(33,47)(34,48)(35,49)(37,52)(38,55)(39,43)(41,42)(46,51)(50,56)(53,54);(2,6)(5,7)(9,13,15,14)(10,12,17,22)(11,20,21,23)(16,19,18,24)(26,33)(27,37)(28,40)(34,43)(38,54)(41,44)(45,49)(51,56);(9,14,15,13)(10,12,17,22)(11,20,21,23)(16,24,18,19)(27,37)(31,35)(36,52)(38,54)(45,49)(46,50)(51,56)(53,55)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893871);