Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1418 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,34)(27,36,38,31)(28,33,40,49)(30,43,42,35)(32,44,48,41)(37,51,53,46)(39,47,55,52)(45,54,50,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,38)(26,31,34,36)(28,37,40,53)(30,39,42,55)(32,45,48,50)(33,46,49,51)(35,47,43,52)(41,54,44,56);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,32)(27,37)(29,40)(30,44)(31,45)(33,43)(34,48)(35,49)(36,50)(38,53)(39,56)(41,42)(46,52)(47,51)(54,55);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28)(26,33)(27,37)(29,40)(30,41)(31,46)(32,43)(34,49)(35,48)(36,51)(38,53)(39,54)(42,44)(45,52)(47,50)(55,56);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(26,34)(27,38)(28,40)(35,43)(39,55)(41,44)(45,50)(46,51)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,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, 7108102);