Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,31,36)(27,32,39,47)(28,33,42,50)(29,34,30,35)(37,45,54,56)(38,46,43,51)(40,48,53,55)(41,49,44,52);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,39)(26,32,36,47)(28,40,42,53)(29,38,30,43)(33,48,50,55)(34,46,35,51)(37,44,54,41)(45,52,56,49);(2,5)(6,8)(25,28,31,42)(26,33,36,50)(27,37,39,54)(29,41,30,44)(32,45,47,56)(34,49,35,52)(38,53,43,40)(46,55,51,48);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,30)(26,34,36,35)(27,38,39,43)(28,41,42,44)(32,46,47,51)(33,49,50,52)(37,53,54,40)(45,55,56,48);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(27,38)(29,30)(32,46)(34,35)(37,53)(39,43)(40,54)(41,44)(45,55)(47,51)(48,56)(49,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6622238);