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 | |
1358 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)(25,26,30,35)(27,31,36,45)(28,33,40,50)(29,32,42,34)(37,44,39,48)(38,46,51,43)(41,47,54,49)(52,56,53,55);(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)(26,31)(28,39)(29,38)(30,36)(32,46)(33,48)(34,43)(35,45)(37,40)(41,53)(42,51)(44,50)(47,55)(49,56)(52,54);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(26,32)(27,36)(28,40)(29,42)(31,43)(33,49)(34,35)(44,56)(45,46)(47,50)(48,55)(52,53);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,28)(26,33)(27,37)(29,41)(30,40)(31,44)(32,47)(34,49)(35,50)(36,39)(38,52)(42,54)(43,55)(45,48)(46,56)(51,53);(10,14)(15,21)(16,22)(23,24)(27,36)(28,40)(31,45)(33,50)(38,51)(41,54)(43,46)(47,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fhi,d^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=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6861275);