Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,15,11,16,29,40)(7,12,19,32,20,33,50,62)(8,21,23,35,24,39,28,13)(9,14,27,38,26,37,55,63)(17,41,43,57,44,61,49,30)(18,45,48,67,47,31,58,60)(22,51,52,36,25,53,54,34)(42,65,66,56,46,64,68,59);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,20)(13,30)(14,31)(15,32)(16,33)(21,41)(22,42)(23,43)(24,44)(25,46)(26,47)(27,48)(28,49)(29,50)(34,56)(35,57)(36,59)(37,45)(38,60)(39,61)(40,62)(51,64)(52,66)(53,65)(54,68)(55,58)(63,67);(2,4)(5,8,11,24)(6,13,16,35)(7,17,20,44)(9,25,26,22)(10,28,29,23)(12,30,33,57)(14,36,37,34)(15,39,40,21)(18,46,47,42)(19,49,50,43)(27,52,55,54)(31,59,45,56)(32,61,62,41)(38,51,63,53)(48,66,58,68)(60,64,67,65);(5,9,10,27,11,26,29,55)(6,14,15,38,16,37,40,63)(7,18,19,48,20,47,50,58)(8,22,23,52,24,25,28,54)(12,31,32,60,33,45,62,67)(13,34,21,51,35,36,39,53)(17,42,43,66,44,46,49,68)(30,56,41,64,57,59,61,65)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373025);