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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,18)(14,17)(15,22)(20,23)(21,24)(25,26)(27,36)(28,40)(29,34)(30,32)(31,38)(33,41)(35,43)(37,45)(39,46)(42,48)(44,52)(47,53)(49,50)(51,54)(55,56);(2,5)(3,7)(9,11,14,22)(10,15,17,19)(12,20,18,24)(13,23,16,21)(25,27)(26,31)(28,37)(29,38)(30,39)(32,44)(33,45)(34,36)(35,46)(40,49)(41,50)(42,51)(43,52)(47,54)(48,55)(53,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,18)(10,16,17,13)(11,20,22,24)(15,21,19,23)(26,32)(28,41)(30,43)(31,44)(33,48)(34,35)(36,46)(37,50)(39,52)(40,47)(45,55)(49,54);(2,5)(6,8)(10,17)(12,18)(15,19)(20,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,47)(34,40)(35,48)(36,49)(38,50)(39,51)(43,53)(44,54)(46,55)(52,56);(1,3,4,7)(2,6,5,8)(9,13,14,16)(10,18,17,12)(11,21,22,23)(15,20,19,24)(26,34)(27,38)(28,41)(32,35)(39,52)(42,53)(45,49)(54,55)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,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, 6871666);