Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,34,29,33,41,47)(27,31,37,44,36,43,51,55)(28,38,40,52,39,32,45,46)(35,48,50,56,49,42,53,54);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,31)(28,35)(29,36)(30,37)(32,42)(33,43)(34,44)(38,48)(39,49)(40,50)(41,51)(45,53)(46,54)(47,55)(52,56);(2,6)(3,7)(25,28)(26,32)(27,35)(29,39)(30,40)(31,42)(33,38)(34,46)(36,49)(37,50)(41,45)(43,48)(44,54)(47,52)(51,53)(55,56);(2,6)(5,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(26,33)(27,36)(28,39)(34,47)(37,51)(40,45)(42,48)(54,56);(10,16)(14,17)(15,24)(22,23)(27,36)(31,43)(35,49)(37,51)(42,48)(44,55)(50,53)(54,56)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315465);