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 | |
1319 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)(25,26,30,35)(27,32,39,48)(28,40,43,33)(29,34,31,36)(37,53,55,46)(38,47,44,51)(41,56,54,49)(42,52,45,50);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,39)(26,32,35,48)(28,41,43,54)(29,38,31,44)(33,49,40,56)(34,47,36,51)(37,45,55,42)(46,52,53,50);(2,5)(6,8)(25,28,30,43)(26,33,35,40)(27,37,39,55)(29,42,31,45)(32,46,48,53)(34,50,36,52)(38,54,44,41)(47,56,51,49);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,31)(26,34,35,36)(27,38,39,44)(28,42,43,45)(32,47,48,51)(33,50,40,52)(37,54,55,41)(46,56,53,49);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(27,38)(29,31)(32,47)(34,36)(37,54)(39,44)(41,55)(42,45)(46,56)(48,51)(49,53)(50,52)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=gi,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, 6619967);