Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1240 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)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,33,30,36,29,35)(27,32,43,45,39,48,38,47)(28,34,44,49,41,51,42,52)(37,46,54,55,40,50,53,56);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,39)(26,32,36,48)(28,40,41,37)(29,38,31,43)(33,45,35,47)(34,50,51,46)(42,54,44,53)(49,56,52,55);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(26,33)(29,31)(32,45)(34,49)(35,36)(38,43)(42,44)(46,55)(47,48)(50,56)(51,52)(53,54);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28)(26,34)(27,37)(29,42)(30,41)(31,44)(32,46)(33,49)(35,52)(36,51)(38,53)(39,40)(43,54)(45,55)(47,56)(48,50);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(28,41)(34,51)(37,40)(42,44)(46,50)(49,52)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=f,d^2=fh,e^2=f,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=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855517);