Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1232 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)(25,26,31,33,30,36,29,35)(27,32,43,48,37,47,39,45)(28,40,44,50,41,34,42,49)(38,52,54,55,51,46,53,56);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27,30,37)(26,32,36,47)(28,38,41,51)(29,39,31,43)(33,48,35,45)(34,46,40,52)(42,53,44,54)(49,56,50,55);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(26,33)(27,37)(28,41)(29,31)(32,45)(34,50)(35,36)(40,49)(46,56)(47,48)(52,55)(53,54);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(25,28)(26,34)(27,38)(29,42)(30,41)(31,44)(32,46)(33,49)(35,50)(36,40)(37,51)(39,53)(43,54)(45,55)(47,52)(48,56);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(27,37)(28,41)(32,47)(34,40)(39,43)(42,44)(45,48)(49,50)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=fhi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6870451);