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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1326 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)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,36)(28,33)(29,44)(30,35)(31,34)(32,40)(37,53)(38,47)(39,48)(41,56)(42,52)(43,51)(45,50)(46,55)(49,54);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,41)(29,38)(30,40)(31,39)(33,49)(34,47)(35,36)(37,42)(43,54)(44,48)(45,55)(46,52)(50,53)(51,56);(2,5)(6,8)(25,28,30,43)(26,33,35,51)(27,37,40,55)(29,42,31,45)(32,46,36,53)(34,50,44,52)(38,54,39,41)(47,49,48,56);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,31)(26,34,35,44)(27,38,40,39)(28,42,43,45)(32,47,36,48)(33,50,51,52)(37,54,55,41)(46,49,53,56);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(27,39)(29,31)(32,47)(34,44)(36,48)(37,41)(38,40)(42,45)(46,49)(50,52)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=f,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, 6619956);