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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1399 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)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26)(27,33)(28,43)(29,35)(30,36)(31,46)(32,37)(34,44)(38,52)(39,48)(40,49)(41,55)(42,50)(45,51)(47,53)(54,56);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27,32,42)(26,33,37,50)(28,38,29,39)(30,40,31,41)(34,47,51,56)(35,48,43,52)(36,49,46,55)(44,53,45,54);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,32,29)(26,34,37,51)(27,38,42,39)(30,44,31,45)(33,47,50,56)(35,46,43,36)(40,53,41,54)(48,55,52,49);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,29,32,28)(26,35,37,43)(27,39,42,38)(30,44,31,45)(33,48,50,52)(34,46,51,36)(40,53,41,54)(47,55,56,49);(10,14)(15,21)(16,22)(23,24)(25,30,32,31)(26,36,37,46)(27,40,42,41)(28,44,29,45)(33,49,50,55)(34,35,51,43)(38,53,39,54)(47,48,56,52)Polycyclic presentation:
2^9|b^2=fi,c^2=fhi,d^2=fhi,e^2=i,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, 6861172);