Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1221 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)(3,6)(4,5)(7,8)(11,17)(14,18)(19,20)(22,23)(25,26,31,44,29,34,30,35)(27,36,45,47,38,32,39,51)(28,40,43,33,41,49,42,48)(37,52,55,50,53,56,54,46);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27,29,38)(26,32,34,36)(28,37,41,53)(30,39,31,45)(33,46,48,50)(35,47,44,51)(40,56,49,52)(42,54,43,55);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(25,28)(26,33)(27,37)(29,41)(30,43)(31,42)(32,46)(34,48)(35,49)(36,50)(38,53)(39,55)(40,44)(45,54)(47,52)(51,56);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(26,34)(27,38)(28,41)(35,44)(39,45)(42,43)(46,50)(52,56);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(27,38)(32,36)(37,53)(39,45)(46,50)(47,51)(52,56)(54,55)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=fhi,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6870492);