Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1317 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)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,34,30,35,43,51)(27,36,38,46,39,50,42,31)(28,32,41,47,40,33,48,49)(37,52,55,45,54,53,56,44);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,39)(26,31,35,46)(28,37,40,54)(29,42,43,38)(32,44,33,45)(34,50,51,36)(41,56,48,55)(47,53,49,52);(2,5)(6,8)(11,17)(14,21)(18,19)(22,23)(25,28,30,40)(26,32,35,33)(27,37,39,54)(29,41,43,48)(31,44,46,45)(34,47,51,49)(36,52,50,53)(38,55,42,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,33)(27,37)(29,41)(30,40)(31,45)(32,35)(34,49)(36,53)(38,55)(39,54)(42,56)(43,48)(44,46)(47,51)(50,52);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(28,40)(32,33)(37,54)(41,48)(44,45)(47,49)(52,53)(55,56)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393211);