Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1325 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,34)(27,31,38,45)(28,32,41,48)(29,33,36,43)(35,42,50,54)(37,44,40,47)(39,46,51,55)(49,53,52,56);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(27,35)(28,39)(29,36)(31,42)(32,46)(33,43)(37,40)(38,50)(41,51)(44,47)(45,54)(48,55);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,36,50,30,38,29,35)(26,31,43,54,34,45,33,42)(28,40,51,49,41,37,39,52)(32,47,55,53,48,44,46,56);(10,16)(14,17)(15,22)(21,23)(28,41)(32,48)(37,40)(39,51)(44,47)(46,55)(49,52)(53,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,28,29,39,30,41,36,51)(26,32,33,46,34,48,43,55)(27,37,35,49,38,40,50,52)(31,44,42,53,45,47,54,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fghi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6743013);