Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1322 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,37)(28,33)(29,44)(30,34)(31,36)(32,39)(35,46)(38,53)(40,49)(41,50)(42,48)(43,51)(45,52)(47,55)(54,56);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,39)(26,32,34,37)(28,40,41,42)(29,31,46,45)(33,48,50,49)(35,36,44,52)(38,54,55,43)(47,56,53,51);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,31,43,30,41,45,54)(26,33,36,51,34,50,52,56)(27,38,29,40,39,55,46,42)(32,47,35,48,37,53,44,49);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(28,41)(32,37)(35,44)(36,52)(38,55)(40,42)(43,54);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,29,45,27,30,46,31,39)(26,35,52,32,34,44,36,37)(28,42,54,55,41,40,43,38)(33,49,56,53,50,48,51,47)Polycyclic presentation:
2^9|b^2=i,c^2=fhi,d^2=fi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6738704);