Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1424 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)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,29,31)(27,35,38,50)(28,40,41,33)(30,44,36,34)(32,46,45,39)(37,53,42,51)(43,49,52,55)(47,54,48,56);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(26,31)(28,41)(32,45)(34,44)(35,50)(37,42)(43,52)(54,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,36,46,29,38,30,39)(26,32,44,50,31,45,34,35)(28,42,52,54,41,37,43,56)(33,48,55,51,40,47,49,53);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(25,28)(26,33)(27,37)(29,41)(30,43)(31,40)(32,47)(34,49)(35,51)(36,52)(38,42)(39,54)(44,55)(45,48)(46,56)(50,53);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(27,38)(28,41)(32,45)(33,40)(35,50)(39,46)(43,52)(49,55)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819482);