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 | |
1392 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)(25,26,32,37)(27,33,41,51)(28,42,29,35)(30,36,31,46)(34,44,53,43)(38,54,39,48)(40,49,45,50)(47,55,52,56);(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)(26,33)(28,39)(29,38)(30,45)(31,40)(32,41)(34,52)(35,54)(36,50)(37,51)(42,48)(43,55)(44,56)(46,49)(47,53);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28)(26,34)(27,38)(29,32)(30,43)(31,44)(33,47)(35,36)(37,53)(39,41)(40,55)(42,46)(45,56)(48,50)(49,54)(51,52);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,29,32,28)(26,35,37,42)(27,39,41,38)(30,43,31,44)(33,48,51,54)(34,46,53,36)(40,55,45,56)(47,49,52,50);(10,14)(15,21)(16,22)(23,24)(25,30,32,31)(26,36,37,46)(27,40,41,45)(28,43,29,44)(33,49,51,50)(34,35,53,42)(38,55,39,56)(47,54,52,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fhi,d^2=fh,e^2=i,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, 6861274);